Prove Rigorously That the Convex combination Homotopy is Continuous in Both Variables

Convex Functions, Partial Orderings, and Statistical Applications

In Mathematics in Science and Engineering, 1992

1.25 Remark

Let U be an open convex set in L, x ₀ ∈ U, y ∈ L, and define the function g(t) = f(x ₀ + ty) where t ∈ (a, b) such that x ₀ + ty ∈ U for all t. The function f: U → $ℝ$ is said to be convex if g(t) is a convex function on (a, b) (Roberts and Varberg, 1973, p. 91).

1.26 Theorem

(a) Let f be convex on an open set U ⊆ L. If f is bounded from above in a neighborhood of a point in U, then f is locally Lipschitz in U, hence Lipschitz on any compact subset of U, and f is continuous on U.

(b) If f is convex on the open set U ⊆ $ℝ$ ⁿ , then f is Lipschitz on every compact subset of U and continuous on U.

Proof

See Roberts and Varberg (1973, pp. 93–94).

1.27 Theorem

(a) Assume that f is defined on the open convex set U ⊆ L. If f is convex on U and Fréchet differentiable at x ₀, then, for x ∈ U,

(1.26) $f(\text{x})-f({\text{x}}_0)\ge f^{\prime }({\text{x}}_0)(\text{x}-{\text{x}}_0)$

holds. If f is differentiable throughout U, then f is convex iff (1.26) holds for all x, x ₀ ∈ U. Furthermore, f is strictly convex iff the inequality is strict.

(b) Let f: U → $ℝ$ be continuous and Fréchet differentiable on the open set U ⊆ L. Then f is (strictly) convex if f′ is (strictly) increasing on U.

(c) Let f be continuously differentiable and suppose that the second derivative exists throughout an open set U ⊆ L. Then f is convex on U iff f″(x) is nonnegative definite for every x ∈ U. Furthermore, if f″(x) is positive definite on U, then f is strictly convex.

Proof

See Roberts and Varberg (1973, pp. 98–101).

1.28 Remarks

(a) We shall say that f′ is increasing on U if for x, y ∈ U we have

$(f^{\prime }(\text{x})-f^{\prime }(y))(\text{x}-y)\ge 0,$

and that f′ is strictly increasing on U if this inequality is strict for all x ≠ y.

(b) A function f: U → M (U ⊆ L; L, M are normed linear vector spaces) is Fréchet differentiable at x ₀ (x ₀ ∈ U) if there exists a linear transformation T: L → M such that

$\underset{\text{x}\to {\text{x}}_0}{\lim }\frac{f(\text{x})-f({\text{x}}_0)-T(\text{x}-{\text{x}}_0)}{\Vert \text{x}-{\text{x}}_0\Vert }=0,$

which is equivalent to

$f(\text{x})=f({\text{x}}_0)+T(\text{x}-{\text{x}}_0)+o\left(\Vert \text{x}-{\text{x}}_0\Vert \right)$

as x → x ₀. The linear transformation T is called the Fréchet derivative and is denoted by f′(x ₀).

(c) A similar derivative of the Fréchet derivative is called the second Fréchet derivative. This derivative is a symmetric bilinear transformation defined on L x L, i.e., f″ _{k, h} (x) = f″ _{h, k} (x) (h, k ∈ L). Note that if f: U → $ℝ$ is continuously differentiable on the open convex set U ⊆ L and f″(x) exists throughout U, then for any x, x ₀ ∈ U there is an s ∈ (0, 1) such that

(1.27) $f(\text{x})=f({\text{x}}_0)+f_h^{\text{'}}({\text{x}}_0)+\frac{1}{2}{f}_{h,h}^{"}({\text{x}}_0+sh),$

where h = x − x ₀.

(d) A symmetric bilinear transformation B(h, k) defined on L x L is positive (nonnegative) definite if for every h ∈ L (h ≠ 0), we have

$B(h,h)>0(B(h,h)\ge 0).$

(e) The following definition is also valid: A (continuous real-valued) function f is operator convex on (λ, v) if f(αa + βb) ≤ αf(a) + βf(b) for positive reals α, β such that α + β = 1 and operators a, b with their spectra in (λ, v). (See Davis, 1957, for a brief survey of operator functions and Ando, 1978, for further comments on classes of operator functions.)

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Computational Theory of Iterative Methods

In Studies in Computational Mathematics, 2007

Definition 8.2.4

Let S⊆ ℝ ⁿ be an open convex set, and let G:S→ ℝ ⁿ be a differentiable operator on S. Let x ₀∈S, and assume:

(8.2.14) $G^{\prime }\left(x_0\right)=I\left(theidentitymatrix\right)$

there exists η ≥ 0 such that

(8.2.15) $\left|\right|G\left(x_0\right)\left|\right|\le \eta ;$

there exists an ℓ₀≥ 0 such that

(8.2.16) $\left|\right|G^{\prime }\left(x\right)-G^{\prime }\left(x_0\right)\left|\right|\le {\ell }_0\left|\right|x-x_0\left|\right|\mathrm{for}\mathrm{all}x\in S.$

Define:

(8.2.17) $h_0=2{\ell }_0\eta .$

We say that G satisfies the weak center-Kantorovich conditions in x ₀ if

(8.2.18) $h_0\le 1.$

We also say that G satisfies the strong center-Kantorovich conditions in x ₀ if

(8.2.19) $h_0\le \frac{c^2}{2}.$

Moreover define

(8.2.20) $r_1=\frac{c-\sqrt{c^2-h_0}}{{\ell }_0},$

(8.2.21) $r_2=\frac{c+\sqrt{c^2-h_0}}{{\ell }_0}$

and

(8.2.22) $R=\left[r_1,r_2\right]\mathrm{for}{\ell }_0\ne 0.$

Furthermore ifℓ₀= 0,define

(8.2.23) $r_1=\frac{\eta }{c}$

and

(8.2.24) $R=\left[r_1,\infty \right).$

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THE FIXED-POINT PRINCIPLE

L.V. KANTOROVICH , G.P. AKILOV , in Functional Analysis (Second Edition), 1982

THEOREM 1.

Let X be a B – space, K a non-empty compact convex set in X, and let f: K → 2 ^K be a many-valued mapping satisfying the conditions:

1)

for each x ∈ K , the set f (x) is a non-empty convex subset of K ;

2)

f is a closed mapping.

Then f has a fixed point.

Proof. ^*As K is compact, Hausdorff's Theorem shows that for each ɛ > 0 there exists a finite ɛ-net for K, consisting of x _ɛ1, x _ɛ2,…, x _{ɛ n (ɛ)}, say. For every x ∈ K, we set

${\phi }_{\epsilon i}\left(x\right)=\max \left(\epsilon -\left|\right|x-x_{\epsilon i}\left|\right|,0\right)\hspace{1em}\hspace{1em}\hspace{1em}(i=1,\mathrm{\dots }n\left(\epsilon \right)).$

Obviously, the ϕ_{ɛ i} are continuous non-negative functions on K. As {x _{ɛ i} } ^{n (ɛ)} _{i = 1}is an ɛ-net, for each x ∈ K there is at least one i with || x – x _{ɛ i} ||<ɛ. Therefore we have ϕ_{ɛ i} (x) > 0 for this i. This argument shows that the following definition is meaningful:

$w_{\epsilon i}\left(x\right)={\phi }_{\epsilon i}\left(x\right)/\underset{j=1}{\overset{n\left(\epsilon \right)}{\Sigma }}{\phi }_{\epsilon j}\left(x\right)\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(i=1,\mathrm{\dots },n\left(\epsilon \right)).$

Now we fix an arbitrary point y _{ɛ i} ∈ f (x _{ɛ i} ) (i = 1,…, n (ɛ)) and define a continuous single-valued mapping f _ɛ: K → K, which acts according to the formula

$f_{\epsilon }\left(x\right)=\underset{i=1}{\overset{n\left(\epsilon \right)}{\Sigma }}{w}_{\epsilon i}\left(x\right)y_{\epsilon i}.$

From the conditions y _{ɛ i} ∈ K, w _{ɛ i} (x) ⩾ 0, ∑ w _{ɛ i} (x) = 1, and the convexity of K it follows that f _ɛ(x) ∈ K. Hence, for any ɛ > 0, we have a continuous single-valued mapping f _ɛ: K → K. By Schauder's Theorem (3.1), this mapping has a fixed point x _ɛ: thus f _ɛ(x _ɛ) = x _ɛ.

As K is compact, there exist a sequence {ɛ _n } of positive numbers and a point x ^*ɛ K such that the following conditions are satisfied:

$\text{a})\underset{n\to \infty }{\lim }{\epsilon }_n=0;\text{b})x_{{\epsilon }_n}\to x*;\text{c})f_{{\epsilon }_n}\left(x_{{\epsilon }_n}\right)=x_n.$

We show that x ^*is the fixed point of f we are seeking. Set U _δ= f (x ^*) + B _δ, where B _δ= {y:|| y ||<δ}, δ > 0. We shall show that x ^*∈ U _δ for every δ > 0, and so, as f (x ^*) is closed, we obtain the required result, that x ^*∈ f (x ^*) (the fact that f (x ^*) is closed follows from the fact that f is a closed mapping).

It is clear that U _δ is an open convex set and that f (x ^*) ⊂ U _δ. Given x ^*and U _δ, we can choose, by Lemma 1, a ball K _ɛ= {x: || x – x ^* ||<ɛ} such that f (K _ɛ) ⊂ U _δ. By conditions a) and b), there exists a number N such that ɛ _n < ɛ/2 and x _{ɛ n} ∈ K _ɛ/2whenever n ⩾ N. If w _{ɛ n i} (x _{ɛ n} ) > 0, then

$\left|\right|x_{{\epsilon }_{n}i}-x_{{\epsilon }_n}\left|\right|<{\epsilon }_n<\epsilon /2$

and

$\left|\right|x_{{\epsilon }_{n}i}-x*\left|\right|\le \left|\right|x_{{\epsilon }_{n}i}-x_{{\epsilon }_n}\left|\right|+\left|\right|x_{{\epsilon }_n}-x*\left|\right|<\epsilon /2+\epsilon \mathrm{/2}=\epsilon .$

Hence, for n > N we have x _{ɛ n i} ∈ K _ɛ, for all i such that w _{ɛ n i} (x _{ɛ n} ) > 0. For these i, we have

(1) $y_{{\epsilon }_{n}i}\in f\left(x_{{\epsilon }_{n}i}\right)\subset f\left(K_{\epsilon }\right)\subset U_{\delta }.$

By condition c), we deduce that

(2) $x_{{\epsilon }_n}=\sum {\omega }_{{\epsilon }_{n}i}\left(x_{{\epsilon }_n}\right)y_{{\epsilon }_{n}i}.$

From (1) and (2) we see that, for n ⩾ N, the point x _{ɛ n} is a convex combination of just those points y _{ɛ n i} that lie in U _δ, and so, as U _δ is convex, we have x _{ɛ n} ∈ U _δ. Taking limits as n → ∞, we see that x ^*∈ $\overline{U}$ _δ⊂ U _2δ. As we have already remarked, it follows from this that x ^*∈ f (x ^*). This completes the proof of Kakutani's Theorem.

5.3. Now we present an application of Kakutani's Theorem to the theory of games.

We consider a game between two people (player I and player II) having zero sum: this means that one player wins by whatever amount the other loses. A player's strategy is a complete enumeration of all moves that this player will make in each of the possible positions in the course of the game. We denote the set of all strategies (or the space of strategies) of player I by X, and the set of strategies of player II by Y.

A pay-off function is a real-valued function K (x, y) defined on X × Y, the number K (x, y) being interpreted as the winnings of player I if player I chooses strategy x and player II strategy y. The number – K (x, y) is interpreted as the winnings of player II in the same situation. A triple (X, Y, K) where X and Y are sets and K (x, y) is a function on X × Y, is called a game.

When player I chooses his strategy x ∈ X, he must assume, knowing nothing about player II's strategy, that his opponent will choose the worst strategy from player I's point of view—that is, that player I's winnings will be $\underset{y\in Y}{\inf }K\left(x,y\right)$ . Hence it is natural for player I to seek a strategy x ₀∈ X such that

(3) $\underset{y\in Y}{\inf }K\left(x_0,y\right)=\underset{x\in X}{\max }\underset{y\in Y}{\inf }K\left(x,y\right).$

Bearing in mind that player II's winnings are – K (x, y), we see from (3) that player II must look for a strategy y ₀∈ Y such that

$\underset{y\in X}{\inf }\left\{-K\left(x,y_0\right)\right\}=\underset{y\in Y}{\max }\underset{x\in X}{\inf }\left[-K\left(x,y\right)\right],$

or, what is the same thing,

(4) $\underset{x\in X}{\sup }K\left(x,y_0\right)=\underset{y\in Y}{\min }\underset{x\in X}{\sup }K\left(x,y\right).$

Comparison of (3) and (4) shows that a necessary and sufficient condition for the existence of strategies x ₀∈ X and y ₀∈ Y satisfying (3) and (4) is that for all x ∈ X, y ∈ Y the following inequality should hold:

(5) $K\left(x,y_0\right)\le K\left(x_0,y_0\right)\le K\left(x_0,y\right),$

and a necessary and sufficient condition for this is that

(6) $\underset{y\in Y}{\min }\underset{x\in X}{\sup }K\left(x,y\right)=\underset{x\in X}{\max }\underset{y\in Y}{\inf }K\left(x,y\right).$

The common value of the two sides of (6) is called the value of the game. Each pair of strategies x ₀, y ₀satisfying (5) is called a solution of the game, and the strategies x ₀, y ₀themselves are called optimal strategies. ^* To prove the existence of a solution for a game one is required to prove a minimax theorem, which consists in verifying equation (6) for the given game. With the help of Kakutani's Theorem, we now prove one very general theorem of this type. First we state the following lemma, whose proof is left to the reader.

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Convex Functions, Partial Orderings, and Statistical Applications

In Mathematics in Science and Engineering, 1992

2.39 Theorem

Let X be a normed space and D ⊂ X be an open convex set. Let the discrete functional A ∈ M ₊ ¹(D, $ℝ$ ) be given by

$\begin{array}{cccc}A(f)={\displaystyle \sum _{k=1}^n{p}_k}f(x_k),& P_n=1,& p_k\ge 0,& k=1,\dots ,n.\end{array}$

If there exists a point a ∈ D with the property

${\displaystyle \sum _{k=1}^n{p}_k}a*(x_k,\alpha -x_k)\ge 0,$

then f(α) ≥ A(f).

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Convex Functions, Partial Orderings, and Statistical Applications

In Mathematics in Science and Engineering, 1992

13.1 Theorem

For n ≥ 1 let ϕ: A → ℝ be a continuous convex function where A ⊂ ℝ ⁿ is an open convex set such that P[X ∈ A] = 1. Then μ ∈ A and

(13.1) $E\varphi (X)={\displaystyle \underset{A}{\int }\varphi }(x)dF(x)\ge \varphi (\mu ).$

Proof

The fact that μ ∈ A is easy to verify. To show that (13.1) holds we use the proof in Marshall and Olkin (1979, p. 454).

For z ∈ A and i = 1, …, n, let

${\varphi }_{(i)}^{+}(z)=\underset{\in \downarrow 0}{\lim ╡}\frac{1}{\in }[\varphi (z_1,\mathrm{\dots },z_{i-1,}{z}_i+\in ,z_{i+1},\mathrm{\dots },z_n)-\varphi (z_1,\mathrm{\dots },z_n)],$

and denote

${\nabla }^{+}\varphi (x)=({\varphi }_{(1)}^{+}(x),\mathrm{\dots },{\varphi }_{(n)}^{+}(x)),\begin{array}{c}x\in A.\end{array}$

Then ∇⁺ϕ(x) is Borel-measurable and, more importantly,

$\varphi (x)-\varphi (z)\ge [{\nabla }^{+}\varphi (z)][x-z),\begin{array}{c}x,z\in A\end{array}$

holds (Marshall and Olkin, 1979, p. 451). Choosing z = μ, we have

(13.2) $\begin{array}{cc}\varphi (x)-\varphi (\mu )\ge [{\nabla }^{+}\varphi (\mu )](x-\mu )& \text{a}\text{.s}\text{.}\end{array}$

The proof follows by taking expectations on both sides of (13.2).

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Analysis of Combinatorial Neural Codes: An Algebraic Approach

Carina Curto , ... Nora Youngs , in Algebraic and Combinatorial Computational Biology, 2019

7.4.1 Convex Realizability

One of the main questions motivating the construction of the neural ideal is to determine which codes have realizations using convex sets, and which do not. Many types of neurons with receptive fields, including both 2D and 3D place cells, have natural convex receptive fields. Structural features of the code are known which either guarantee or prohibit the existence of a realization with convex sets, and the neural ideal and canonical form can often be used to detect these features. We will refer to a code which has a realization using open convex receptive fields in ${\mathbb{R}}^d$ for some d as a convex code.

As Exercise 7.8 shows, the simplicial complex alone is insufficient to characterize a code. In addition to the fact that different codes may have the same simplicial complex, codes with the same simplicial complex may have very different properties with respect to convex realizability.

Exercise 7.37

Let ${\mathcal{C}}_1,{\mathcal{C}}_2$ , and ${\mathcal{C}}_3$ be the three codes from Exercise 7.8. Show that ${\mathcal{C}}_1$ has a realization with convex open sets in $\mathbb{R}$ , that ${\mathcal{C}}_2$ does not have such a realization, but can be realized with convex open sets in ${\mathbb{R}}^2$ , and that ${\mathcal{C}}_3$ cannot be realized with convex open sets in ${\mathbb{R}}^d$ for any dimension d.

As seen in Exercise 7.37 and previously in Exercise 7.5, there exist codes which are not convex. The obstructions to convexity in these exercises are topological in nature, and are specific examples of a broad class of obstructions known as local obstructions, defined in [13, 14]. These obstructions follow from an application of the Nerve lemma.

Lemma 7.1 (Nerve Lemma)

If the sets $\mathcal{U}=\{U_1\dots ,U_n\}$ are convex, then the homotopy type of ${\bigcup }_{i=1}^n{U}_i$ is equal to the homotopy type of the nerve $\mathcal{N}(\mathcal{U})$ . In particular, ${\bigcup }_{i=1}^n{U}_i$ and $N(\mathcal{U})$ have exactly the same homology groups.

The Nerve lemma is a consequence of [15, Corollary 4G.3]. Local obstructions arise in instances where features of the code dictate that any realization of the code using convex sets would violate the Nerve lemma; thus, a code which can be realized with convex sets must have no local obstructions.

Determining if a code $\mathcal{C}$ has local obstructions to convexity can be reduced to the question of determining whether $\mathcal{C}$ contains a particular minimal code, as Theorem 7.2 will show. However, we first require a definition.

Definition 7.11

Let Δ be a simplicial complex. For any σ ∈ Δ, the link of σ in Δ is

$\begin{array}{l}\hfill {\text{Lk}}_{\sigma }(\Delta )\stackrel{\text{def}}{=}\{\omega \in \Delta |\sigma \cap \omega =\varnothing \text{and}\sigma \cup \omega \in \Delta \}.\end{array}$

Example 7.15

Consider the simplicial complex

$\begin{array}{l}\hfill \Delta =\{\varnothing ,\{1\},\{2\},\{3\},\{4\},\{1,2\},\{1,3\},\{1,4\},\{2,3\},\{3,4\},\{1,2,3\}\}.\end{array}$

Then, Lk_{1}(Δ) = {∅, {2}, {3}, {4}, {2, 3}}, whereas Lk_{2}(Δ) = {∅, {1}, {3}, {1, 3}} and Lk_{{1, 3}}(Δ) = {∅, {2}} (See Fig. 7.4).

Fig. 7.4. The simplicial complex Δ from Example 7.15 (at left) and the link Lk_{1}(Δ) (at right). Here, Lk_{1}(Δ) is disconnected, and thus not contractible.

The necessary condition for a code to have no local obstructions will involve determining whether links are contractible. Informally, a topological space (such as a simplicial complex) is contractible if it is connected and has no "holes"; more formally, a space is contractible if it is homotopy-equivalent to a single point. In the previous example, Lk_{1}(Δ) is disconnected, and thus is not contractible, but the other two links are both contractible.

Theorem 7.2

(Theorem 1.3 from [13])

For each simplicial complex Δ, there is a unique minimal code ${\mathcal{C}}_{min}(\Delta )$ with the following properties:

1.

the simplicial complex of ${\mathcal{C}}_{min}(\Delta )$ is Δ, and

2.

any code $\mathcal{C}$ satisfying $\Delta (\mathcal{C})=\Delta$ has no local obstructions if and only if $\mathcal{C}\supseteq {\mathcal{C}}_{min}(\Delta )$ .

Determining this minimal code depends only on the simplicial complex $\Delta (\mathcal{C})$ :

$\begin{array}{l}\hfill {\mathcal{C}}_{min}(\Delta )=\{\sigma \in \Delta |{\text{Lk}}_{\sigma }(\Delta )\text{is noncontractible}\}\cup \{\varnothing \}.\end{array}$

Exercise 7.38

Recall code $\mathcal{C}={\mathcal{C}}_3$ from Exercises 7.8 and 7.37. With $\Delta =\Delta (\mathcal{C})$ as computed in that exercise, show that Lk_{{1, 2}}(Δ) is not contractible. Conclude that $1100\in {\mathcal{C}}_{min}$ , and that since $\mathcal{C}$ does not contain 1100, it cannot be convex.

Exercise 7.39

Recall $\mathcal{C}=\{000,010,001,110,101\}$ , the code from Exercise 7.5. Find $\Delta (\mathcal{C})$ . Then, identify a set σ such that Lk _σ (Δ) is not contractible, and the associated codeword c with supp(c) = σ is not in $\mathcal{C}$ . Conclude that $\mathcal{C}$ is not convex.

While it can be difficult in general to detect these local obstructions from the canonical form, it is sometimes possible. Exercise 7.38 exhibits a simple example of such a situation.

Example 7.16

The code ${\mathcal{C}}_3$ from Exercises 7.8 and 7.38 has canonical form

$\begin{array}{l}\hfill CF(J_{{\mathcal{C}}_3})=\{x_3{x}_4,x_1{x}_2(1-x_3)(1-x_4),x_3(1-x_1)(1-x_2),x_4(1-x_2),x_4(1-x_1)\}.\end{array}$

The first two polynomials together tell us that $(U_1\cap U_2)\subset (U_3\cup U_4)$ and $U_3\cap U_4=\varnothing$ . This gives us a clue that we may have an obstruction using σ = {1, 2}, since in any convex realization, $U_1\cap U_2$ should also be convex and thus connected. However, we just found that it is contained in the union of two disjoint sets, so it cannot be connected.

We can show that examples such as this one provide a pattern for similar local obstructions.

Exercise 7.40

Suppose $\mathcal{C}\subset {\mathbb{F}}_2^n$ . Show that if there is some set $\sigma \subset [n]$ such that both of the following conditions hold:

1.

$x_{\sigma }(1-x_i)(1-x_j)\in CF(J_{\mathcal{C}})$

2.

$x_{\sigma }{x}_i{x}_j\in J_{\mathcal{C}}$

then $\mathcal{C}$ is not convex, as we can use σ to find a local obstruction.

Demonstrating that a code has a local obstruction proves immediately that the code cannot be convex. However, the converse does not hold; there are codes which have no local obstructions but still have no open convex realization, as in the following example.

Example 7.17

(Lienkaemper et al. [16])

The following code on five neurons has no local obstructions, but has no open convex realization:

$\begin{array}{ll}\hfill C=& \{00000,00100,00010,10100,10010,01100,00110,00011,11100,10110,\hfill \\ \hfill & 10011,01111\}.\hfill \end{array}$

The obstruction to convexity in this case is more geometric in its flavor—one can use convexity show that the straight line which connects points in two different regions has only a limited set of possible regions it can pass through, and only in certain orders, and this can be shown to be impossible in all cases.

Thus, local obstructions can only prove nonconvexity; we cannot assume that a code without local obstructions is necessarily convex. That said, several large classes of codes are known to have convex realizations. We will now show some examples of these classes, and give algebraic signatures from the canonical form which can tell us immediately if a code fits into one of these classes. The first such class is simplicial complexes.

Theorem 7.3

(Curto et al. [13])

If $\mathcal{C}$ is a simplicial complex, then $\mathcal{C}$ has a convex realization.

Proposition 7.2

Let $\mathcal{C}$ be a neural code. Then $\mathcal{C}$ is a simplicial complex if and only if $CF(J_{\mathcal{C}})$ consists only of monomials.

Exercise 7.41

Prove Proposition 7.2.

Theorem 7.3 and Proposition 7.2 combined show that by computing the canonical form $CF(J_{\mathcal{C}})$ , we can immediately detect if $\mathcal{C}$ is a simplicial complex, and if so, conclude that $\mathcal{C}$ is convex. However, it is quite specific to require that a code be a simplicial complex, and many codes which are quite clearly realizable (including any code where one receptive field is covered by others, as in Example 7.1) are not simplicial complexes. Fortunately, there is a relaxation of the simplicial complex condition which is also known to be convex.

Definition 7.12

A code $\mathcal{C}$ is intersection-complete if for any $\mathbf{c},\mathbf{d}\in \mathcal{C}$ , there is a codeword $\mathbf{v}\in \mathcal{C}$ such that $\mathrm{supp}(\mathbf{v})=\mathrm{supp}(\mathbf{c})\cap \mathrm{supp}(d)$ .

Exercise 7.42

Show that codes which are simplicial complexes are intersection-complete. Give an example to show that the converse does not hold.

The larger class of intersection-complete codes is also known to be convex realizable.

Theorem 7.4

(Cruz et al. [17])

If $\mathcal{C}$ is intersection-complete, then $\mathcal{C}$ has a convex realization.

Intersection-complete codes are more general than simplicial complex codes, but we can use the canonical form to detect them as well, as the following result indicates.

Theorem 7.5

(Curto et al. [18])

A code $\mathcal{C}$ containing the all-zero word 0 is intersection-complete if and only if $CF(J_{\mathcal{C}})$ consists only of monomials, and mixed monomials of the form

$\begin{array}{l}\hfill \prod _{i\in \sigma }{x}_i(1-x_j).\end{array}$

From the previous two results, we see that codes whose canonical forms are entirely monomial (as with simplicial complexes) or allow only the most basic mixed monomials (intersection-complete codes) are convex. This might lead us to suspect that having mostly monomial-like relationships is somehow necessary for convexity. However, this is not the case, as codes with no monomials at all in their canonical form are also convex.

Theorem 7.6

(Curto et al. [13])

If $\mathcal{C}$ is a neural code and $CF(J_{\mathcal{C}})$ contains no monomials, then $\mathcal{C}$ has a convex realization.

Furthermore, codes with no monomial relationships are also known to have convex realizations in very low dimensions (1 or 2); see Fig. 7.5 and related discussion. This property also has a simple characterization in terms of the canonical form, as seen in the following exercise.

Fig. 7.5. Constructing a 2D realization of $\mathcal{C}=\{000,100,001,101,110,111\}$ . The codeword 111 is placed at the center of a polygon inscribed within a circle; the remaining codewords are assigned to the spaces created around the edges. We then take U _i to be the union of the regions labeled by codewords with neuron i firing.

Exercise 7.43

Show that $CF(J_{\mathcal{C}})$ contains no monomials if and only if $\mathcal{C}$ contains the codeword 1 = 111…1.

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Introduction to Ulam stability theory

In Ulam Stability of Operators, 2018

4 Other functional equations and inequalities in several variables

Now, we present examples of stability results that have been obtained for various functional equations and inequalities in several variables.

Let us start with a result of D.H. Hyers and S.M. Ulam [78] for convex functions.

Theorem 14

Let $D\subset {ℝ}^n$ be an open convex set with non-empty interior and a function $f:D\to ℝ$ satisfy

$f\left(\mathit{tx}+\left(1-t\right)y\right)\le \mathit{tf}\left(x\right)+\left(1-t\right)f\left(y\right)+\epsilon ,x,y\in D,0\le t\le 1.$

Let B be a closed bounded convex subset of D. Then there exists a convex function $g:B\to ℝ$ such that

$\mid f\left(x\right)-g\left(x\right)\mid \le k_n\epsilon ,x\in B,$

where

$k_n=\frac{n^2+3n}{4n+4}.$

Next, let us recall a result of Z. Kominek [93] for the Jensen equation.

Theorem 15

Let $D\subset {ℝ}^n$ be a bounded set with a nonempty interior and Y be a Banach space. Assume that there exists x ₀ ∈ D such that D ₀ := D − x ₀ satisfies the condition

$\frac{1}{2}{D}_0\subset D_0.$

Then, for each function f : D → Y such that

${\displaystyle \underset{x,y\in D}{sup}}‖f\left(\frac{x+y}{2}\right)-\frac{f\left(x\right)+f\left(y\right)}{2}‖<\infty ,$

there exists $g:{ℝ}^n\to \mathrm{Y}$ such that

$g\left(\frac{x+y}{2}\right)=\frac{g\left(x\right)+g\left(y\right)}{2},x,y\in {ℝ}^n,$

and

${\displaystyle \underset{x\in D}{sup}}\Vert f\left(x\right)-g\left(x\right)\Vert <\infty .$

The first author treating Hyers-Ulam stability of the quadratic equation was F. Skof [138], who proved the following.

Theorem 16

Let X be a normed vector space and Y a Banach space. If a function f : X → Y fulfills

$\Vert f\left(x+y\right)+f\left(x-y\right)-2f\left(x\right)-2f\left(y\right)\Vert \le \delta ,x,y\in X,$

for some δ > 0, then for every x ∈ X the limit

$g\left(x\right)={\displaystyle \underset{n\to \infty }{lim}}\frac{f\left(2^{n}x\right)}{2^{2n}}$

exists and g is the unique solution of the functional equation

$g\left(x+y\right)+g\left(x-y\right)=2g\left(x\right)+2g\left(y\right),x,y\in X,$

with

$\Vert f\left(x\right)-g\left(x\right)\Vert \le \frac{\delta }{2},x\in X.$

Since then, the stability problem for the quadratic equation has been extensively investigated by a number of mathematicians in, e.g., [56, 80, 124, 127].

The Hyers-Ulam-Rassias stability of the pexiderized versions of the additive, Jensen, and quadratic equations, i.e., of the following three equations

$\begin{array}{c}\hfill f\left(x+y\right)=g\left(x\right)+h\left(y\right),\hfill \\ \hfill f\left(\frac{x+y}{2}\right)=\frac{g\left(x\right)+h\left(y\right)}{2},\hfill \\ \hfill f\left(x+y\right)+g\left(x-y\right)=h\left(x\right)+k\left(y\right),\hfill \end{array}$

has been studied in [79, 95, 96]. In particular the subsequent theorem has been proved in [96].

Theorem 17

Let X be a normed space, Y be a Banach space, and let f, g, h : X → Y be mappings. Assume that K ≥ 0 and p ≠ 1 are real numbers with

$\Vert f\left(x+y\right)-g\left(x\right)-h\left(y\right)\Vert \le K\left(\Vert x\Vert {}^p+\Vert y\Vert {}^p\right),x,y\in X\backslash \left\{0\right\}.$

Then there exists a unique additive mapping T : X → Y such that

$\Vert f\left(x\right)-T\left(x\right)-f\left(0\right)\Vert \le \frac{4K\left(3+3^p\right)}{2^p\mid 3-3^p\mid }\Vert x\Vert {}^p,x\in X\backslash \left\{0\right\}.$

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Information Geometry

Jun Zhang , Ting-Kam Leonard Wong , in Handbook of Statistics, 2021

3.2 λ-Deformed Legendre duality

Legendre duality has been generalized to c-duality which is used extensively in the theory of optimal transport (Villani, 2003, 2008). In our context, the bilinear pairing x ⋅ y in (26) is replaced by c(x ⋅ y), where c is monotone but non-linear in general. In particular, we take c(t) = κ _λ (t). By Theorem 1, the requirement of convexity of f(x) is replaced by the requirement of convexity of g(x) = γ _λ (f(x)) on a convex set Ω ∋ x, and the λ-conjugate variable is given by the λ-deformed Legendre transformation. Following Wong and Zhang (2021), we made the following definition.

Definition 5

λ-conjugate function

Fix $\lambda \in \mathbb{R},\lambda \ne 0$ . Given a function $f:\mathrm{\Omega }\to \mathbb{R}$ on some $\mathrm{\Omega }\subset {\mathbb{R}}^d$ , we define f ^(λ) on ${\mathrm{\Omega }}^{\prime }\subset {\mathbb{R}}^d$ by

(30) $\begin{array}{l}\hfill f^{(\lambda )}(u)=\underset{x\in \mathrm{\Omega }}{sup}\left({\kappa }_{\lambda }(x\cdot u)-f(x)\right),u\in {\mathbb{R}}^d.\end{array}$

We call f ^(λ) the λ-conjugate of f.

The following result connects the notion of λ-convexity and the deformed duality under the function κ _λ .

Theorem 1 λ-duality

Let λ ≠ 0 and let f be a smooth and λ-convex function on some open convex set $\mathrm{\Omega }\subset {\mathbb{R}}^d$ . Further assume 1 − λ∇f(x) ⋅ x > 0 on Ω. ^c

(i)

Consider f ^(λ) as a function on some other open convex set Ω′. Then the operation of λ-conjugation is involutive on Ω: (f ^(λ))^(λ) = f;

(ii)

For x ∈ Ω, define the λ-gradient ∇ ^λ by

(31) $\begin{array}{l}\hfill {\nabla }^{\lambda }f(x):=\frac{1}{1-\lambda \nabla f(x)\cdot x}\nabla f(x).\end{array}$

Then u = ∇ ^λ f(x) is a C ¹ diffeomorphism from Ω onto its range Ω′ which is an open set;

(iii)

For x ∈ Ω and u = ∇ ^λ f(x) we have 1 + λx ⋅ u > 0 and the following identity holds

(32) $\begin{array}{l}\hfill f(x)+f^{(\lambda )}(u)\equiv \frac{1}{\lambda }log(1+\lambda x\cdot u)={\kappa }_{\lambda }(x\cdot u).\end{array}$

In reminiscent of Legendre duality, we call the above set of relations the λ -deformed Legendre duality, or λ -duality in short.

Proof

The proof of this theorem is given in scattered parts in Wong (2018, Section 3.3). Here we just rephrased the results in terms of the λ-duality.

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Barrels and Other Features of TVS's

Eric Schechter , in Handbook of Analysis and Its Foundations, 1997

BARRELS AND ULTRABARRELS

27.20.

Remarks Ultrabarrels are a generalization of barrels. Barrels are simpler to define, but they are mainly useful in locally convex spaces; ultrabarrels can be useful in the more general setting of topological vector spaces. The theories of barrels in LCS and ultrabarrels in TVS are closely analogous; the analogy will be developed in the sections below.

The definitions of barrels and ultrabarrels involve absorbing sets (defined in 12.8). In a TVS, absorbing sets may be viewed as "generalized neighborhoods of 0" — any neighborhood of 0 is absorbing, but not every absorbing set is a neighborhood of 0. For instance, sketch a graph of {(x, y) ∈ ℝ² : |y| ≥ x ² or y = 0}; show that this set is absorbing but is not a neighborhood of (0, 0) when ℝ² is equipped with its usual topology.

27.21.

Definition Let X be a topological vector space. A barrel in X is a subset of X that is closed, convex, balanced, and absorbing. (Those terms are defined in 5.13, 12.3, and 12.8.)

27.22.

Basic properties of barrels Let X be a topological vector space.

a.

If X is a locally convex space, then X has a neighborhood base at 0 consisting of barrels.

b.

If ρ is a continuous seminorm on X and k > 0, then {x ∈ X : ρ(x) ≤ k} is a barrel.

c.

If B is a barrel in X, then its Minkowski functional μ _B is a seminorm on X. Moreover, μ _B is continuous if and only if B is a neighborhood of 0.

27.23.

Definition Let X be a vector space. A string in X is a sequence of sets (S _n : n ∈ ℕ) that are balanced, absorbing, and satisfy S _n ⊇ S _n+1 + S _n+1 for all n. The S _n 's are then called the knots of the string.

In a topological vector space, a closed string is a string whose knots are closed sets; those closed sets are called ultrabarrels. We still have a string if we discard the first few knots of a string; hence every ultrabarrel may also be viewed as the first knot of a closed string.

In a topological vector space, a neighborhood string is a string, all of whose knots are neighborhoods of 0. (Some mathematicians call this a topological string.)

27.24.

Basic properties of strings and ultrabarrels

a.

If B is a barrel in a TVS X, then B is also an ultrabarrel — it is a knot of the closed convex string (S _n ) defined by S _n = 2⁻ⁿ B.

b.

If (S _n ) is a string in a vector space X, then {S _n } forms a neighborhood base at 0 for a TVS topology on X.

Conversely, if X is a TVS, then X has a neighborhood base at 0 consisting of ultrabarrels.

c.

If ρ is an F-seminorm on a vector space X and k is a positive constant, then the sets S _n = {x ∈ X : ρ(x) ≤ 2⁻ⁿ } form a string.

If ρ is a continuous F-seminorm on a TVS X, then the sequence (S _n ) defined as above is a closed string; thus its members are ultrabarrels.

d.

If (S _n ) is a string in a vector space X, then there exist an F-seminorm ρ on X and positive numbers a _n , b _n decreasing to 0 that satisfy

$\begin{array}{lllll}\left\{x\in :\rho \left(x\right)\le a_n\right\}\hfill & \subseteq \hfill & S_n\hfill & \subseteq \hfill & \left\{x\in X:\rho \left(x\right)\le b_n\right\}\hfill \end{array}$

for all n. (Hint: The sets V _n = {(x, y) ∈ X × X : x − y ∈ S _2n } satisfy the hypotheses of 4.44.)

Suppose, moreover, that X is a TVS. Then ρ is continuous if and only if (S _n ) is a neighborhood string. (Hint: An F-seminorm is continuous if and only if it is continuous at 0.)

e.

If (S _n ) and (T _n ) are strings and S _n + T _n = U _n , then (U _n ) is a string.

27.25.

Proposition Suppose X is a complete metric space or, more generally, a Baire space. Then:

(i)

If X is a TVS, then X is ultrabarrelled, as defined in 27.26(U1).

(ii)

If X is an LCS, then X is barrelled, as defined in 27.27(B1).

Proof It suffices to prove (i), since that result implies (ii) by 27.24.a. Let (S _n ) be a closed string; we wish to show that S ₁ is a neighborhood of 0. Since S ₂ is absorbing, $X={\displaystyle {\cup }_{k=1}^{\infty }k{S}_2}$ . By 20.15(B), some kS ₂ has nonempty interior. Hence S ₂ has nonempty interior. Say x ₀ ∈ int(S ₂). Thus x ₀ + G ⊆ S ₂ where G is some neighborhood of 0. Since S ₂ is symmetric, we have 0 + (G − G) = (x ₀ + G) - (x ₀ + G) ⊆ S ₂ − S ₂ = S ₂ + S ₂ ⊆ S ₁.

27.26.

Theorem and definition Let (X, J) be a real or complex topological vector space. Then the following conditions on (X, J) are equivalent. If any, hence all, of these conditions are satisfied, we say (X, J) is ultrabarrelled.

(U1)

Every ultrabarrel in X is a neighborhood of 0.

(U2)

(F-Seminorms Property.) Each lower semicontinuous F-seminorm on X is continuous.

(U3)

(Banach's Closed Graph Property.) Let Y be an F-space. Let f : X → Y be linear and have closed graph. Then f is continuous.

(U4)

(Neumann's Nonlinear Closed Graph Property.) Let Y be a locally full F-space. Let Ω ⊆ X be an open convex set. Suppose f : Ω → Y is a convex operator whose graph is a closed subset of Ω − Y. Then f is continuous.

(U5)

(Banach-Steinhaus Uniform Boundedness Property.) Let Y be a topological vector space. Let Φ be a collection of continuous linear maps from X into Y that is toplinearly bounded pointwise. Then Φ is equicontinuous.

(U6)

(Neumann's Nonlinear Uniform Boundedness Property.) Let Y be an ordered topological vector space that is locally full. Let Ω ⊆ X be an open convex set. Let Φ be a collection of continuous convex maps from Ω into Y. Suppose Φ is toplinearly bounded pointwise. Then Φ is equicontinuous.

Proof of this theorem begins in Section 27.31.

27.27.

Theorem and Definition Let (X, J) be a real or complex locally convex space. Then the following conditions on (X, J) are equivalent. If any, hence all, of these conditions are satisfied, we say (X, J) is barrelled.

(B1)

Every barrel in X is a neighborhood of 0.

(B2)

(Seminorms Property.) Each lower semicontinuous seminorm on X is continuous.

(B3)

(Closed Graph Property.) Let Y be a Fréchet space. Let f : X → Y be linear and have closed graph. Then f is continuous.

(B4)

(Neumann's Nonlinear Closed Graph Property.) Let Y be a locally full Fréchet space. Let Ω ⊆ X be an open convex set. Suppose f : Ω → Y is a convex function, whose graph is a closed subset of Ω x Y. Then f is continuous.

(B5)

(Uniform Boundedness Property.) Let Y be a locally convex space. Let Φ be a collection of continuous linear maps from X into Y that is toplinearly bounded pointwise. Then Φ is equicontinuous.

(B6)

(Neumann's Nonlinear Uniform Boundedness Property.) Let Y be a locally full, locally convex space. Let Ω ⊆ X be an open convex set. Let Φ be a collection of continuous convex maps from Ω into Y that is toplinearly bounded pointwise. Then Φ is equicontinuous.

Proof of this theorem begins in Section 27.31.

Remark A seventh characterization of barrelled spaces will be given in 28.30.

27.28.

Corollaries: classical versions Let X be an F-space. By 27.25:

a.

Any lower semicontinuous F-seminorm on X is continuous.

b.

Uniform Boundedness Theorem. If Y is a topological vector space and Φ is a collection of continuous linear maps from X into Y such that Φ(x) = {f(x) : f ∈ Φ} is a bounded subset of Y for each x ∈ X, then Φ is equicontinuous.

c.

Closed Graph Theorem. If Y is an F-space and f : X → Y is a linear map whose graph is a closed subset of X − Y, then f is continuous.

d.

If Ω is an open convex subset of X and f : Ω → ℝ is a convex function whose graph is a closed subset of Ω − ℝ, then f is continuous.

27.29.

Example application Let (Ω, S, μ) be a measure space, let X be a Banach space, and let α, β ∈ (0, +∞) be exponents such that L ^α(μ, X) ⊆ L ^β(μ, X). Then the inclusion $L^{\alpha }\left(\mu ,X\right)\stackrel{\subseteq }{\to }{L}^{\beta }\left(\mu ,X\right)$ is continuous.

Proof It suffices to show that the inclusion map has closed graph. Suppose f _n → f in L ^α(μ, X) and f _n → g in L ^β(μ, X); we are to show that f = g. By 22.31(ii) we may pass to subsequences such that f _n → f and f _n → g pointwise μ-almost everywhere.

Remarks This example is taken from Villani [1985]. That paper also shows the following interesting result: Let X be a Banach space, let (Ω, S, μ) be a measure space, and let α, β ∈ (0, +∞) with α > β. Then

$\begin{array}{lll}{L}^{\alpha }\left(\mu ,X\right)\subseteq L^{\beta }\left(\mu ,X\right)\hfill & \text{if\hspace{0.17em}and\hspace{0.17em}only\hspace{0.17em}if}\hfill & \text{inf}\left\{\mu \left(S\right):S\in S,\mu \left(S\right)>0\right\}>0;\hfill \\ L^{\alpha }\left(\mu ,X\right)\supseteq L^{\beta }\left(\mu ,X\right)\hfill & \text{if\hspace{0.17em}and\hspace{0.17em}only\hspace{0.17em}if}\hfill & \text{sup}\left\{\mu \left(S\right):S\in S,\mu \left(S\right)<0\right\}<\infty ;\hfill \end{array}$

Special cases of this were given in 22.34. (Villani's paper only shows this for X = ℝ, but that case easily yields the general case since all the functions in L ^α(μ, X) or L ^β(μ, X) are measurable, and we can separate the "regular" condition from the "not too big" condition — see the remarks in 22.28.)

27.30.

Change of scalar field Let X be a complex topological vector space (respectively, a complex locally convex space). Then X, with the same topology, may also be viewed as a real topological vector space (respectively, a real locally convex space) if we "forget" how to multiply members of X by nonreal scalars. Let us denote these two TVS's by X _ℂ and X _ℝ. Note that the choice of scalars affects the definitions of "balanced" and "absorbing;" hence it affects the definitions of "barrel" and "ultrabarrel." Show that

a.

If B is an (ultra)barrel in X _ℂ, then B is also an (ultra)barrel in X _ℝ. Likewise, any (F−)seminorm on X _C is also an (F−)seminorm on X _ℝ.

Hence, if X _ℝ is (ultra)barrelled, then X _ℂ is (ultra)barrelled, too.

b.

It is possible for X _ℝ to have more (ultra)barrels than X _ℂ.

For instance, show that the set B = {z ∈ ℂ : |Re(z)| ≤ 1 and |Im(z)| ≤ 2} is both a barrel and an ultrabarrel in X _ℝ, but is neither in X _ℂ since B is not balanced in X _ℂ.

c.

Nevertheless, X _ℝ is (ultra)barrelled if and only if X _ℂ is (ultra)barrelled.

For the moment, we shall prove this equivalence using only definitions (U2) and (B2); proofs with the other definitions in 27.26 and 27.27 will follow from the arguments given in the next subchapter.

We have already established half of this "if and only if" result. Now assume X _ℂ is (ultra)barrelled — i.e., it satisfies condition (U2) or (B2). To show the same for X _ℝ, let ρ be any lower semicontinuous (F-)seminorm on X _ℝ; we wish to show that ρ is continuous on X _ℝ. Note that X _ℝ and X _ℂ differ only in their algebraic operations — they are the same set, and they have the same topology; so a function is continuous on X _ℝ if and only if it is continuous on X _ℂ. Define γ : X → [0, +∞) as in 26.5.b. As we noted in 26.5.b, this function γ is also lower semicontinuous on X, and γ is an (F-)seminorm on X _ℂ. Hence, by our assumption, γ is continuous. From the inequality ρ ≤ γ, we see that ρ is continuous at 0. Since ρ is a G-seminorm, we have |ρ(u) - ρ(v)| ≤ ρ(u − v), and therefore ρ is continuous.

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Probabilities and Potential C

In North-Holland Mathematics Studies, 1988

Convex Cones with a Compact Base

62

It will be convenient here to suppose that K is contained in a closed affine hyperplane H ∈ E, which doesn't pass through the origin. We then denote by C the cone with base K, in other words C = U_{t≥0} tK.

If μ is a positive bounded measure on K we define as in no. 29 the resultant r(μ) of μ.

In many applications in analysis it is the cone C which arises naturally, the choice of K - the intersection of C with a hyperplane H = {h = 1}-corresponding to a certain way of normalizing the elements of C, which is somewhat arbitrary. See the examples at the end of the chapter.

Since C is a convex cone, there corresponds to it an order relation compatible with addition, which we will denote < in this section. We will consider it only on C itself,

(62.1) $\left(\text{x}\le \text{y}\right)\iff \left(\text{y}-\text{x}\in \text{C}\right)\left(\text{x},\text{y}\in \text{C}\right)$

Remark that we have tx ≤ × for 0 ≤ t ≤ 1, and × ≤ tx for t ≥ 1. On the other hand if we represent the hyperplane in the form {h = 1} where h is continuous linear on E, h is increasing on C, and takes the value 0 on C only at the point 0. There is a bisection between the decompositions of × ∈ K in C,

$\text{x}={\text{y}}_1+\mathrm{\dots }+{\text{y}}_{\text{n}};{\text{y}}_1,\mathrm{\dots .},{\text{y}}_{\text{n}}\in \text{C},{\text{y}}_{\text{i}}\ne 0\text{f}\text{o}\text{r}\text{a}\text{l}\text{l}\text{i}$

and the barycentric representations in K

$\text{x}={\text{t}}_1{\text{z}}_1+\mathrm{\dots }+{\text{t}}_{\text{n}}{\text{z}}_{\text{n}};{\text{z}}_1,\mathrm{\dots },{\text{z}}_{\text{n}}\in \text{k},{\text{t}}_{\text{i}}>0\text{f}\text{o}\text{r}\text{a}\text{l}\text{l}\text{i},{\sum }_{\text{i}}{\text{t}}_{\text{i}}=1$

the correspondence being given by t_i = h(y_i), z_i = y_i/t_i. For example to say that × is extremal amounts to saying that in every decomposition × = Σ_iy_i of × into elements of C, all the y_i are proportional to x.

The main tool will be the following lemma, which permits calculation of $⊣$ $\hat{\text{f}}$ by means of decompositions in the cone C:

63

THEOREM. Let f be a continuous function on K, which we extend (without changing notation) as a positively homogeneous function on C. Then we have for all × ∈ K

(63.1) $\hat{\text{f}}\left(\text{x}\right)=\text{sup}{\sum }_{\text{i}}\text{f}\left({\text{y}}_{\text{i}}\right)$

the sup being taken over all the finite decompositions × = Σ_iy_i, y_i ∈ C.

Proof. Thanks to the relation between decompositions and barycentric representations, this formula can be written $\hat{\text{f}}\left(\text{x}\right)=\text{s}\text{u}{\text{p}}_{\text{μ}}\text{μ}\left(\text{f}\right)$ , μ running over the set of probability laws on K with barycentre x, and with finite support. We know moreover (40) that $\hat{\text{f}}\left(\text{x}\right)=\text{s}\text{u}\text{p}\text{μ}\left(\text{f}\right)$ , μ running over set of all the laws with barycentre x. It thus suffices to be able to approximate each law μ, in the sense of vague convergence, by discrete measures with the same barycentre . To establish this consider a finite covering of K by convex open sets ω _i; it is easy to construct a representation of the law μ as a convex combination μ = σ_it_iμ_i, where for each i the law μ_i is carried by ω_i. The measure with finite support Σ_it_iε_b(μi) = λ thus admits the same barycentre as μ. On the other hand, as E is locally convex and K compact, there exist coverings of the preceding type finer than any given covering, and the corresponding measures λ are arbitrarily close to μ for narrow convergence.

Here is Choquet's uniqueness theorem (the proof says a little more than the statement). Recall that each point of K is the barycentre of at least one maximal measure for the order ⊣_λ (41), and that the maximal measures are exactly those carried by the boundary when K is metrizable (43).

64

THEOREM. The two following properties are equivalent

1)

The cone C is a lattice for its own order ≤.

2)

Every point of K is the barycente of a unique maximal law.

Proof. The implication 2) ⇒ 1) is the easy part: if there is uniqueness, the application μ ↦ r(μ) is an isomorphism of the convex cone of bounded maximal measures to the cone C, and the cone of maximal measures is a lattice (in the metrizable case, it is the cone of measures on the boundary, in the non-metrizable case, it is characterized by the property $\text{μ}\left(\hat{\text{f}}-\text{f}\right)=0$ for f ∈ $\mathcal{C}$ (K)). Conversely suppose that C is a lattice, we shall establish not only property 2) of the statement but many other interesting properties.. Here is the first:

(64.1) $\underset{¯}{\text{For}\text{all}\text{continuous}\text{convex}}\text{f}\underset{¯}{\text{on}}\text{k},\hat{\text{f}}\underset{¯}{\text{is}\text{affine}\text{on}}\text{K}.$

As $⊣$ $\hat{\text{f}}$ is concave, it suffices to show that $⊣$ $\hat{\text{f}}$ is also convex. Extending f and $⊣$ $\hat{\text{f}}$ from K to C as positively homogeneous functions (without changing notationcf. 63), this comes down to showing that $\hat{\text{f}}\left(\text{x}+\text{y}\right)\le \hat{\text{f}}\left(\text{x}\right)+\hat{\text{f}}\left(\text{y}\right)$ for x, y ∈ C. But by 63

$\hat{\text{f}}\left(\text{x}+\text{y}\right)=\text{sup}{\sum }_{\text{i}}\text{f}\left({\text{z}}_{\text{i}}\right),\text{where}\text{x}+\text{y}\text{=}{\sum }_{\text{i}}{\text{z}}_{\text{i}}.$

As C is a lattice, the classical decomposition lemma asserts the existence of decompositions z_i = x_i + y_i with Σ_ix_i = x, Σ_iy_i = y. As f is convex, we have f(z_i) ≤ f(x_i) + f(y_i), and hence

${\text{Σ}}_{\text{i}}\text{f}({\text{z}}_{\text{i}})\le {\text{Σ}}_{\text{i}}\text{f}({\text{x}}_{\text{i}})+{\text{Σ}}_{\text{i}}\text{f}({\text{y}}_{\text{i}})\le \hat{\text{f}}(\text{x})+\hat{\text{f}}(\text{y}),$

the required inequality.

To establish the decomposition lemma, we reduce the problem to the case of a decomposition z ▭ z₁ + z₂ with two elements. A proof appears in Bourbaki, Intégr. chap II, §1 no. 1 (reference not guaranteed, the "definitive" edition not yet being ready). But the reader could also extract it from IX.88.

Now, since $⊣$ $\hat{\text{f}}$ is u.s.c. affine, it is strongly affine (remark in no. 53), and we have $\text{μ}\left(\hat{\text{f}}\right)=\hat{\text{f}}\left(\text{x}\right)$ for every law μ with barycentre x. If further μ is is maximal, we have $\text{μ}\left(\hat{\text{f}}\right)=\text{μ}\left(\text{f}\right)$ , so

(64.2) $\text{μ}\left(\text{f}\right)=\hat{\text{f}}\left(\text{x}\right)\text{for}\text{f}\text{continuous convex}\text{.}$

Thus the linear functional μ is determined on the cone V of continuous convex functions. As V-V is dense in $\mathcal{C}$ (K) (Stone-Weierstrass theorem) this uniquely determines μ, and the theorem is proven.

We note a further property in the same vein. It is a consequence of(64.2) that

(64.3) $\text{T}\text{h}\text{e}\text{m}\text{a}\text{p}\text{f}\mapsto \stackrel{⌢}{\text{f}}\text{i}\text{s}\text{a}\text{d}\text{d}\text{i}\text{c}\text{t}\text{i}\text{v}\text{e}\text{o}\text{n}\text{V}.$

Conversely, if this property holds, the map $\text{f}\mapsto \hat{\text{f}}\left(\text{x}\right)$ is a positive linear functional on V-V, which extends to a positive linear functional μ on $\mathcal{C}$ (K), and it is easy to see that every law λ with barycentre × satisfies λ⊣_λ μ: so μ is the unique maximal sweeping of ε_x.

REMARKS.

a)

We say that K is a simplex if every point of E is the barycentre of a unique maximal law, by analogy with the case of $\mathcal{R}$ ⁿ where the word has its usual meaning. We note that the properties (64.1), (64.3) are internal characterizations of simplexes: no recourse is needed to the cone C and its order.

b)

Property (64.1) generalizes to infinite dimensions the remark well known to café waiters: the cups are upset less often when the tray is placed on a three legged pedestal than when it is placed on a table with four unequal legs.

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Source: https://www.sciencedirect.com/topics/mathematics/convex-open-set

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