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A standard name of a strongly extremal point of a convex set

I need to name somehow points $x$ of a bounded convex set $C$ in a Banach space $X$ such that the set $$\{x^*\in X^*:x^*(x)=\max x^*[C]\}$$ of support functionals at $x$ has non-empty interior in the ...
Taras Banakh's user avatar
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Conjugate of composition in Bochner spaces

Let $H$ be a separable Hilbert space (of non-zero dimension), let $(\Omega,\Sigma,\mu)$ be a finite measure space, and let $L^2(\mu;H)$ be the Bochner-space $\mu$-integrable $H$-valued functions. ...
Catologist_who_flies_on_Monday's user avatar
0 votes
4 answers
444 views

Confining a polytope to one side of an affine hyperplane

Judging whether one convex polytope is inside of another when both are expressed as a system of linear inequalities seems not to be an easy problem. This answer on math.stackexchange.com claims the ...
Hans's user avatar
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Let $A,B,C$ be centrally-symmetric convex bodies. What is this function $G(x,y) := \sup_{b \in B}\inf_{a \in A} a^T x - b^T y + \|a-b\|_C$?

Let $A$, $B$, and $C$ be centrally-symmeric convex bodies in $\mathbb R^n$. Note that any such set can such set induces a norm $\|\cdot\|_C$ on $\mathbb R^n$ defined by $\|x\|_C := \sup_{c \in C}c^\...
dohmatob's user avatar
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0 answers
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Proximity operator of lower semi-continuous and convex functions pre-composed with norm

Let $\phi:(-\infty,\infty) \rightarrow (-\infty,\infty]$ be a some-where finite, lower semi-continuous, increasing, and convex function. It is easy to verify that the function $\Phi:\mathbb{R}^n\...
Bernard_Karkanidis's user avatar
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68 views

Can convex functions on product space be approximated by product of convex functions?

I am working on a problem where I need the following property that I guess should be true but I am not able to prove it. I have a bounded convex function $F(x, y)$ on $X\times Y$ (Think of $X=Y=\...
Raghav's user avatar
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1 answer
247 views

When does strict inclusion holds for the domain of subdifferential?

Recall that, given an extended real-valued function $f: \mathbb{R}^n \to (-\infty, \infty]$ Its effective domain is, $$\text{dom}(f) = \{x \in \mathbb{R}^n : f(x) < +\infty\}$$ The subdifferential ...
Shamisen Expert's user avatar
1 vote
0 answers
88 views

Differential of the gradient of a strictly convex function

For $n\geq 2$, we consider $\mathbb{R}^n$ endowed with the usual scalar product. Let $f\in\mathcal{C}^2(\mathbb{R}^n,\mathbb{R})$ be a striclty convex function such that $\nabla f$ is nowhere ...
G. Panel's user avatar
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23 votes
2 answers
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Are such functions differentiable?

In my recent researches, I encountered functions $f$ satisfying the following functional inequality: $$ (*)\; f(x)\geq f(y)(1+x-y) \; ; \; x,y\in \mathbb{R}. $$ Since $f$ is convex (because $\...
M.H.Hooshmand's user avatar
2 votes
1 answer
329 views

Normal cones and KKT conditions

I'm trying to understand a statement from the book "Perturbation Analysis of Optimization Problems", by Bonnans and Shapiro. Let me start by providing some context. In page 148, the authors ...
Ariel Serranoni's user avatar
4 votes
2 answers
178 views

Existence of a fundamental domain for the convex hull of group action on a rational polytope

Let $P \subset \mathbb R^n$ be a compact rational polytope (the affine space spanned by $P$ may not be $\mathbb R^n$). Let $G \subset {\bf GL}(n,\mathbb Z)$ be an arithmetic group. Let $$C = {\rm Conv}...
Li Yutong's user avatar
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1 answer
525 views

A concave function as supremum of upper semi continuous is upper semi continuous

We define a affine(concave), upper semi continuous function and bounded function $f:X \to \mathbb{R}$, where $X \subset \mathbb{R}^{k}$ is compact and convex set. Assume that $T$ is an affine and ...
Adam's user avatar
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1 answer
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One question about how to get $\inf_{\nu\in\mathcal{M}(Z)} \sum_{j=0}^N F_j^*(\nu)=-(\sum_{j=0}^N F_j^*)^*(0)$?

In this paper: Matching for teams by G. Carlier and I. Ekeland (2010), https://www.jstor.org/stable/25619994?seq=1#metadata_info_tab_contents: they claim that follows a standard argument in Ekeland ...
Hermi's user avatar
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8 votes
1 answer
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Probability of a deviation when Jensen’s inequality is almost tight

This is a cross-post to a yet unanswered question in Math StackExchange https://math.stackexchange.com/questions/3906767/probability-of-a-deviation-when-jensen-s-inequality-is-almost-tight Let $X>0$...
Luis L.'s user avatar
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2 answers
942 views

Distance to a closed set. Is this result known?

Given a closed set $\varnothing\neq E\subset\mathbb{R}^n$, let $\operatorname{Unp}(E)$ be the set if points $x\in\mathbb{R}^n$ for which there is a unique point $y\in E$ nearest to $x$. Clearly $E\...
Piotr Hajlasz's user avatar
3 votes
1 answer
361 views

Concavity near the boundary of the distance function

I was reading the paper Quelques remarques sur les problemes elliptiques quasilineaires du second ordre, P. L. Lions, Journal d’Analyse Mathématique volume 45, pages 234–254(1985) and on page 251 he ...
Sean's user avatar
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13 votes
2 answers
654 views

Regularity of convex sets in $\mathbb{R}^n$

The following result is Proposition 2.4.3 in [1]: Theorem. Let $K\subset\mathbb{R}^n$ be a bounded convex set with the non-empty interior. Then $\partial K\in C^{1,1}$ if and only if there is $r>0$...
Piotr Hajlasz's user avatar
0 votes
1 answer
339 views

Is it true that every uniformly continuous strictly convex function on convex compact subset of a finite-dim normed vector space has unique minimizer? [closed]

Let $C$ be a convex compact subset of a finite-dimensional normed vector space and let $f:C \to \mathbb R$ be strictly convex and uniformly continuous function (e.g it is sufficient that $f$ be ...
dohmatob's user avatar
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1 vote
0 answers
99 views

On the differentiability of max-type functions

If $f(x, y)$ is strongly concave in $y$, then from Danskin's theorem we know that $\max_{y\in Y} f(x, y)$ is differentiable if $Y$ is convex and bounded. What if $Y$ is not bounded? Is $\max_{y\in Y} ...
Guojun Zhang's user avatar
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1 answer
105 views

If $C_1\subseteq C_2$ are two closed convex cones that are pointed with $\partial C_1\subseteq \partial C_2$ then is $C_1=C_2$?

Let $C_1$ and $C_2$ be two proper full dimensional closed convex cones in $\mathbb{R}^n$ that are pointed. Suppose that $C_1\subseteq C_2$ and that the boundary of $C_1$ is contained in the boundary ...
slack tide's user avatar
10 votes
4 answers
3k views

Convexity and Lipschitz continuity

It is probably an easy question, but somehow I am stuck. Question Is the following statement true? If yes, how to prove it? Suppose that $f\in C^1(\mathbb{R}^n)$ is convex and $$ \langle\nabla f(x)-\...
Piotr Hajlasz's user avatar
3 votes
3 answers
215 views

Example of a (strictly) proper scoring rule on a general measurable space?

Most of the literature on scoring rules that I know of deals with discrete measurable spaces, but in this paper by Gneiting and Raferty a very general definition of a scoring rule is given. I don't ...
aduh's user avatar
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4 votes
0 answers
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Elementary functions (growing faster than exponential ones) with elementary Legendre–Fenchel transforms

Let $F$ be the set of all convex functions $f\colon[0,\infty)\to[0,\infty)$ with $f(0)=0=f'_+(0)$ and $f_+(\infty-)=\infty$, where $f'_+$ is the right derivative of $f$. For any function $f\in F$, its ...
Iosif Pinelis's user avatar
2 votes
0 answers
98 views

A strong duality for convex functional optimization that admits Lipschitz continuity constraints?

Problem Statement I am looking for formal proof---hopefully textbook material---of two items: an analogue to Slater's condition [1] that obtains strong duality for optimization of convex functionals; ...
Kyle Treleaven's user avatar
1 vote
1 answer
136 views

Find this reference or an alternative where I can find this result

I need this reference, but I couldn't find it online as a PDF. Any help please? J. Sun, X, Zhang, The fixed point theorem of convex-power condensing operator and applications to abstract semilinear ...
Motaka's user avatar
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0 votes
1 answer
605 views

Linear combination of convex functions is constant

Let $\Phi : \mathbb{R}_{++}\to \mathbb{R}$ be a convex function (not necessarily differentiable). Fix an $\alpha \in (0,1)$ and define $$g(t) = \alpha t .\Phi\left(\frac{1}{t} + 1\right) + 2(1-\alpha)...
De vinci's user avatar
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4 votes
0 answers
252 views

Can this function be minimized?

Let $X$ be a locally convex TVS, and let $A$ and $B$ be convex and compact subsets of $X$ with $A \subset B$. Let $f: A \times B \to [0,\infty]$ have the following properties: (1) For all $b \in B$, $...
aduh's user avatar
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1 vote
0 answers
68 views

Fundamental regions in convex programming

In linear programming, the fundamental regions are polyhedra, since those are the intersection of half-spaces defined by linear inequalities. In semidefinite programming, the fundamental regions are ...
Turbo's user avatar
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1 vote
0 answers
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Conditions on triangle inequality for integral kernel

Consider $\int_RK(x,y)f(y)dy$, where $K(x,y) \in M_+(R^2)$. Let $L(t,s)$ be an iterated rearrangement of $K$. Let also $$ A(t,v)=\int_0^{1/v}L(1/t,s)ds, $$ which is decreasing with $v$ and ...
user124297's user avatar
1 vote
1 answer
188 views

Is this $(\Bbb R^{n \times n})^n \to \Bbb R$ function convex?

Let $W := (W_1, W_2,\dots, W_n)$, where $W_i \in \Bbb R^{n \times n}$. Let $x$ be a constant vector. Is the following function convex? $$f(W) := x^TW_1^TW_2^T \cdots W_n^TW_n \cdots W_2W_1x $$
Qiuhai Zeng's user avatar
1 vote
0 answers
240 views

Möbius function and polynomials

Let $\mu$ be the Möbius function. It is well known that $\sum_{n|k} \mu(n) = 0$ for $k>1$. What could be said about the polynomials $R_k = \sum_{n|k} \mu(n) x^n$ for $x \in [0,1]$? There does not ...
A413's user avatar
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0 votes
1 answer
204 views

the subdifferential at points of differentiability in infinite dimensional space

Let $ f: X\to (-\infty,+\infty]$ that $ X$ is an infinite dimensional space. What are the conditions for $f$ and space $X$ to have the following equality correct? $$\partial f(x)=\{\nabla f(x)\}$$ for ...
SS342's user avatar
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7 votes
2 answers
547 views

Is a function of several variables convex near a local minimum when the derivatives are non-degenerate?

This is a cross-post. Let $U \subseteq \mathbb R^n$ be an open subset, and let $f:U \to \mathbb R$ be smooth. Suppose that $x \in U$ is a strict local minimum point of $f$. Let $df^k(x):(\mathbb R^n)^...
Asaf Shachar's user avatar
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4 votes
1 answer
289 views

Variance of random variable decreasing in parameter

I did quite a few numerical computations and think the following is true, but I cannot prove it: Let $\varphi(x):=\sum_{i=1}^n \varphi_i(x_i)$ where $x=(x_1,...,x_n) \in \mathbb{R}^n$ and $\varphi_i \...
Sascha's user avatar
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4 votes
5 answers
616 views

Elementary inequality generalizing convexity of a function on a segment

I am looking for a proof of the following statement which is known to be true as far as I heard. Let $g\colon [a,b]\to \mathbb{R}$ be a smooth function. Assume that $$b-a< \pi.$$ Assume also $$g(a)\...
asv's user avatar
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1 vote
1 answer
445 views

Convexity at a point and Jensen inequality

I am looking for a reference for the following claim: Let $\phi:\mathbb (a,b) \to \mathbb R$ be a continuous function, and let $c \in (a,b)$ be fixed. Suppose that "$\phi$ is convex at $c$". ...
Asaf Shachar's user avatar
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1 vote
0 answers
40 views

Minimax theorems in nonconvex setting

Let $X$ be a topological space, $Z$ be a compact convex subset of $\mathbb R^m$, and let $f:X \times Z \to \mathbb R$ be a continuous function (w.r.t the product topology on $X \times Z$). Question. ...
dohmatob's user avatar
  • 6,716
2 votes
1 answer
334 views

Lower bound on $L^2$ norm of a strongly convex function

Let $f\colon[0, 1] \to \mathbb R$ be an $m$-strongly convex function and $\mu$ be a probability measure on $[0,1].$ For any $t<1$, the goal is to find a lower bound on $\int_{0}^t f^2(x) d\mu(x)$ ...
Statguy's user avatar
  • 23
5 votes
1 answer
362 views

Proving equivalence of two definitions of a convex-type Hamming distance

Update: If somebody can answer my question there, then I will be able to fully answer my question here. Consider $n\in\mathbb N$ and a non-empty set $M\subset\{0,1\}^n$. I have the following ...
Maximilian Janisch's user avatar
2 votes
1 answer
356 views

Concavity of entropy difference

Suppose that $\mathrm{A}$ is a $n\times n$ random matrix with a given distribution. Suppose that $\mathrm{U}$ is a diagonal unitary random matrix, defined as \begin{align*} \begin{bmatrix} \exp(i\...
Mini's user avatar
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5 votes
0 answers
133 views

Conv A = Dual B

I have two cones $A$ and $B$ in a Euclidean space. I want to show that $\mathrm{Conv}\,A=\mathrm{Dual}\,B$; that is, $a\in\mathrm{Conv}\,A$ if and only if $\langle a,b\rangle\ge 0$ for any $b\in B$. ...
Anton Petrunin's user avatar
1 vote
1 answer
74 views

How to compare the minimums of two discrete convex functions?

I have a question that troubled me for a long time. If I have two convex discrete function $f(·)$ and $g(·)$ such that $f(·) \ge g(·)$. (may be not necessary?) Let $x_1 = \text{argmin } f(·)$, ...
Kurt. Z's user avatar
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11 votes
1 answer
502 views

A function with unexpectedly simple Legendre transformation

Let $I(x) = \frac{1}{2\pi} \int_{-2}^2 \sqrt{4-y^2}\ln|x-y|dy$. Then $I(x)$ is a concave function and \begin{equation} I(x)= \begin{cases} \frac{1}{4}x^2-\frac{1}{2}, &\text{if } |x|\leq2 \\ \...
minhtoan's user avatar
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1 vote
1 answer
184 views

On examples of subspaces of $C(X)$ for which state spaces are Choquet Simplices

Let $C(X)$ be the Banach space of all Real valued continuous functions on a compact Hausdorff space $X$. What are examples of uniformly closed subspace $\mathcal{A}$ of $C(X)$ such that $\mathcal{A}$ ...
Tanmoy Paul's user avatar
-1 votes
1 answer
182 views

Dense linear span implies closed convex hull has non-empty interior

Let $X$ be a Fréchet space and let $Y\subseteq X$ such that $\overline{\operatorname{span}(Y)}=X$. It seems intuitive to me that $\operatorname{int}\big(\overline{\operatorname{co}(Y)}\big)$ is a non-...
ABIM's user avatar
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1 vote
0 answers
92 views

strict convexity of the Legendre-Fenchel transform

Let $d$ be a positive integer. Let $L:\mathbb{R}^d\to\mathbb{R}$ be a differentiable function with continuous derivatives. Assume that the Legendre-Fenchel transform of $L$ exists everywhere, is ...
Man Ray's user avatar
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4 votes
1 answer
231 views

Is there a non-convex function with non-decreasing average rate of change?

$\newcommand{\R}{\mathbb R}$ Let $f$ be a function from $\R$ to $\R$. It is said that $f$ is midpoint-convex if for any real $x$ and $y$ we have $f(x-y)+f(x+y)\ge2f(x)$ or, equivalently, $f(x+y)-f(x)\...
Iosif Pinelis's user avatar
0 votes
0 answers
52 views

Separability of Minkowski Sum of well-behaved sets

Let $A$ and $C$ be non-empty simply connected and connected subsets of $\mathbb{R}^k$ and suppose that $C$ is convex. Then is the Minkowski sum $A+C$ separable?
ABIM's user avatar
  • 5,019
2 votes
1 answer
969 views

$\arg\max$ in the dual norm of the nuclear norm

Given a matrix $X \in \mathbb{R}^{m \times n},$ then the spectral norm is defined by $$\left \| X\right\| := \max\limits_{i \in \{1, \dots, \min\{m,n\}\} }\sigma_i (X)$$ whereas the nuclear norm is ...
Santiago Armstrong's user avatar
3 votes
0 answers
46 views

A complete metric space with some convex-type property

Let $(X,d)$ be a complete metric space with this property: for each $x∈X$, $r>0$ and $y∈X$ with $d(x,y)<r$, there exists $z∈X$ such that $d(x,y)+d(y,z)=d(x,z)=r$. I want to know if the family ...
M. Reza. K's user avatar

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