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How many Tverberg partition are in cloud of points?

Tverberg's Theorem: A collection of $(d+1)(r-1) +1$ points in $\mathbb{R}^d$ can always be partitioned into $r$ parts whose convex hulls intersect. For example, $d=2$, $r=3$, 7 points: Let $p_1, p_2,...
D. S.'s user avatar
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3 votes
0 answers
200 views
+100

What's the number of facets of a $d$-dimensional cyclic polytope?

A face of a convex polytope $P$ is defined as $P$ itself, or a subset of $P$ of the form $P\cap h$, where $h$ is a hyperplane such that $P$ is fully contained in one of the closed half-spaces ...
D.S.'s user avatar
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4 votes
1 answer
292 views

An arrangement of hyperplanes [closed]

An arrangement of hyperplanes in $\mathbb{R}^d$ is simple if the hyperplanes are in general position (for every $1\leq k\leq d+1$, the intersection of $k$ hyperplanes is $(d-k)$-dimensional). My ...
D. S.'s user avatar
  • 189
4 votes
0 answers
239 views

Maximum and concavity of function

Let \begin{align} G(x_1,x_2,x_3)=\int_0^{2\pi}\int_0^{2\pi}\int_0^{2\pi} \frac{d\theta_1 d\theta_2 d\theta_3}{(2\pi)^3} p(\vec{x},\vec{\theta}) \log\left( p(\vec{x}, \vec{\theta})\right) \end{...
nervxxx's user avatar
  • 209
6 votes
1 answer
192 views

Determinantal inequality for difference of substochastic matrices

Let $A=(A_{ij})_{1\le i,j\le n}$ be a square matrix with nonnegative real entries. Recall that $A$ is called a substochastic matrix if $$ \forall i,\ \ \sum_j A_{ij}\le 1\ . $$ In the course of my ...
Abdelmalek Abdesselam's user avatar
2 votes
0 answers
68 views

Reference request: books on convex analysis / geometry

I am interested in convex geometry and analysis, especially in its connections with high dimensional probability theory. I was reading the book by Pisier, The volume of convex bodies and Banach space ...
Drew Brady's user avatar
2 votes
1 answer
293 views

Concavity of hypergeometric function ratio

I would like to show that the function, $$ f(x) = \frac{{}_2\mathrm{F}_1\big(\frac{1}{2},\frac{1}{2};c+1\,;x\big)}{{}_2\mathrm{F}_1\big(\frac{1}{2},\frac{1}{2};c\,;x\big)} $$ is concave for $0 < x &...
japalmer's user avatar
  • 321
1 vote
1 answer
39 views

Does uniform convergence of suitable functions yield pathwise convergence of their convex envelopes?

For each $k\ge 1$, let $f_k:\mathbb R\to\mathbb R_+$ be $1-$Lipschitz, increasing such that $f_k(x)\ge x^+$ for $x\in\mathbb R$, $f_k(-\infty)=0$ and $$\lim_{x\to+\infty} \big(f_k(x)-x\big)=0.$$ ...
GJC20's user avatar
  • 1,274
0 votes
1 answer
141 views

When are infimal convolutions contractions?

Let $X$ be a separable Fréchet space and $\varphi,\psi:X\to \mathbb{R}$ be a lower semi-continuous and convex function with $\psi$ bounded below and coercive. Consider the infimal convolution $$ \...
Math_Newbie's user avatar
1 vote
1 answer
79 views

Why are symmetric convex bodies with a smooth boundary and non-vanishing Gaussian curvature of particular interest in harmonic analysis?

I don't work in harmonic analysis or convex analysis, but in some literature of harmonic analysis, I often see the assumption that "let $K$ be a symmetric convex body with a smooth boundary and ...
taylor's user avatar
  • 445
1 vote
0 answers
90 views

Distance between two convex sets

Setting If $A$ an $B$ are two symmetric matrices, we denote by $A >B$ when the matrice $A-B$ is definite positive. In $\left(\mathbb{R}^{*}_{+} \right)^4$, consider the convex set $$ \Lambda = \...
Anthony's user avatar
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5 votes
3 answers
549 views

An inequality in an Euclidean space

For $n\geq 1$, endow $\mathbb{R}^n$ with the usual scalar product. Let $u=(1,1,\dots,1)\in\mathbb{R}^n$, $v\in {]0,+\infty[^n}$ and denote by $p_{u^\perp}$ and $p_{v^\perp}$ the orthographic ...
G. Panel's user avatar
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2 votes
1 answer
174 views

Prékopa-Leindler style inequality?

Does anyone know a simple proof of the following Prékopa-Leindler style inequality: If we have $f_1,f_2,g_1,g_2$ strictly positive functions on $\mathbb{R}$ such that, for any $x_1,x_2 \in \mathbb{R}$,...
Anthony's user avatar
  • 125
0 votes
1 answer
48 views

Do separable cubic constraint and separable quartic constraint SOCP presentable?

I am an engineer who is doing some network modeling and optimization. During my work, I was running into a case that is quite strange. The problem that I am trying to solve seems to be convex and it ...
Tuong Nguyen Minh's user avatar
3 votes
0 answers
161 views

Interchange limit and supremum of functionals over a bounded convex set

Let $(H, \langle\cdot,\cdot\rangle)$ be a separable real Hilbert space and $B\subset H$ be (nonempty) convex and bounded, and suppose that $(\alpha_k)\subset H$ is a sequence for which the limit $\...
fsp-b's user avatar
  • 461
7 votes
2 answers
327 views

Integral means vs infinite convex combinations

Let $(X,\mathcal A, \mu)$ be a probability space, $\mathbb E$ a Banach space, and $f:X\to\mathbb E$ a Bochner integrable function. Does there exist a sequence $(x_k)_{k\ge 1} $ in $X$, and a ...
Pietro Majer's user avatar
  • 58.1k
0 votes
0 answers
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Verifying the Cauchy behavior of a sequence

Let me consider the iteration $x_{n+1}=Tx_n$ and $T$ is a self-map from a non-empty subset $K$ of a smooth Banach space $X$ to itself, satisfying $W(Tx, Ty) \leq W(x, y)$, where $W(x, y)=\Vert x \Vert^...
PPB's user avatar
  • 75
7 votes
3 answers
650 views

A continuous version of Carathéodory's convex hull theorem

A well-known theorem of Caratheodory states that any point in the convex hull of a set $X\subset R^n$ lies in the convex hull of at most $n+1$ points of $X$. I am wondering about a version of this ...
Mohammad Ghomi's user avatar
0 votes
1 answer
84 views

Optimality condition for strongly convex function under sparsity constraint

Let $f: \mathbb{R}^p \to \mathbb{R}$ be a $2s$-sparse strongly smooth, $2s$-sparse strongly convex and twice differentiable function. In other words, there exists positive constants $\alpha, L >0$ ...
De vinci's user avatar
  • 329
2 votes
1 answer
121 views

Is it true that the set $\{x\colon \partial\phi(x)\subset B_r(0)\}$ is convex when $\phi$ is convex?

Let $\phi\colon \mathbb{R}^n\to \mathbb{R}$ be convex. Is it true that the sets $$ A_r = \{x\in \mathbb{R}^n\;\colon\; \partial \phi(x) \subset B_r(0)\},\quad r>0, $$ are convex? For $n=1$ this ...
user522653's user avatar
1 vote
1 answer
133 views

Projection of an element of the $n$-simplex onto subset

Let $\mathbb{S}^{n}$ denote the $n$-dimensional probability simplex and let $\{e_1,...,e_{n+1}\}$ be the canonical basis of $\mathbb{R}^{n+1}$. Consider the subset $\mathbb{S}^{n}(K) \subset \mathbb{S}...
aureliano_buendia's user avatar
0 votes
1 answer
136 views

Property of $p$-norm in the $n$-simplex

Let $\mathbb{S}^{n}$ be the canonical simplex of $\mathbb{R}^{n}$ and let $u = (1/n,\dotsc,1/n)$. Is it true that $$\lVert x - u \rVert_p \leq \lVert y - u \rVert_p$$ implies that $$\lVert x\rVert_p \...
aureliano_buendia's user avatar
5 votes
2 answers
434 views

Extending a convex function to a higher dimensional domain

Let $D\subset{\mathbb R}^2$ be the unit disk, and $I=(-1,1)$. Let $v\in C^2(\bar I)$ be a convex function. Does there exist a convex function $u\in C^2(\bar D)$, such that on the one hand $u(\cdot,0)=...
Denis Serre's user avatar
  • 51.9k
1 vote
1 answer
158 views

Derivative of Moreau envelope in Hilbert space with respect to regularization parameter $\lambda$ using Hopf-Lax formula?

Let $\mathcal H$ be a Hilbert space, $f \colon \mathcal H \to (- \infty, \infty]$ a proper, convex, lower semi-continuous function and $\lambda > 0$. The $\lambda$-Moreau envelope of $f$ is $$ f_{\...
ViktorStein's user avatar
2 votes
0 answers
53 views

Instances of c-concavity outside of optimal transport?

Let $X$ and $Y$ be metric spaces, and let $c:X\times Y\rightarrow \mathbb{R}$ be a nonnegative function which we refer to as a cost. For any $\phi:X\rightarrow \mathbb{R}$ and $\psi:Y\rightarrow \...
Brendan Mallery's user avatar
0 votes
0 answers
74 views

Is a bounded measurable convex function above its interior lower semi-continuous convex envelope?

Let $E$ be a locally convex topological vector space, let $C$ be a convex set which matches the closure of its relative interior $\mathring C=\{ x\in C : \forall y\in C,\exists z\in C,~x\in\mathopen]y,...
P. Quinton's user avatar
0 votes
0 answers
38 views

For a convex body $K\subset R^n$, does the quantity $\min_{t>0} E[t^{-1} \|tg - \pi_K( t g)\|_2]$ have a name? Where has it been studied?

Consider a convex compact set $K\subset R^n$ (with non-empty interior if that helps). Let $\Pi_K:R^n\to K$ be the projection onto $K$, defined as $$ \Pi_K(x) = \operatorname{argmin}_{k\in K} \|k-x\| $$...
jlewk's user avatar
  • 1,574
-1 votes
1 answer
72 views

Regions when a concave function is smaller than another concave function

Let $f_1,f_2:[0,1]\mapsto\mathbb{R}$ be two bounded and concave functions. Assume $f_1(0)<f_2(0)$ and $f_1(1)<f_2(1)$. I want to investigate the set $\mathcal{X}\triangleq\{x\in[0,1]: f_1(x)>...
Eggplant's user avatar
1 vote
1 answer
125 views

Is a bounded convex function $g$ that is non-negative on this particular convex set also non-negative on the its closure?

Setup : Let $S$ be the simplex on $\mathbb N$, i.e. the set of probability distribution on the natural numbers. Suppose we have $p\in S$ such that, for all $n\geq 1$, $p_n > 0$. For any $\emptyset\...
P. Quinton's user avatar
1 vote
1 answer
117 views

Properties of the relatively bounded probability distributions on the simplex over the natural numbers

Setup : Let $S$ be the simplex over $\mathbb N$, i.e. the set of probability distribution over $\mathbb N$. Let $p\in S$ be such that $0 < p_n$ for all $n\in \mathbb N$. Let $H$ be the Hilbert ...
P. Quinton's user avatar
1 vote
1 answer
189 views

Does an uncountable convex combination of elements of a set lie in the convex hull of the set in finite dimension?

Suppose that $\mathcal{F}$ is a finite-dimensional vector space and that $C\subseteq\mathcal{F}$ is a convex subset of $\mathcal{F}$. Is it true that an uncountable convex-combination of elements of $...
rick's user avatar
  • 179
3 votes
0 answers
60 views

Convex combination of cyclically monotone sets

I want to show the following statement, but I am not sure how. Proposition(?): Let $C \in \mathbb{R}^d$ be a compact convex set, and let $u, v : C \to \mathbb{R}$ be smooth convex functions. Suppose $$...
Paruru's user avatar
  • 51
0 votes
0 answers
97 views

Primal optimal attained implies dual optimal attained

Given some optimization problem $$\min_{x \in S \subset \mathbb{R}^n} f_0(x) \quad \text{s.t.} \quad f_i(x) \leq 0, \quad 1\leq i\leq m$$ we can find the dual problem $$\max_{\lambda\in\mathbb{R}^m} g(...
wsz_fantasy's user avatar
1 vote
1 answer
132 views

On the additive property of the subdifferential of lower semicontinuous functions

Let $f:\mathbb R\to\mathbb R$ be a lower semicontinuous function, we define the Fréchet subdifferential of $f$ at $x\in\mathbb R$ by $$\partial^F f(x):=\left\{L\in\mathbb R: \liminf_{v\to0}\frac{f(x+v)...
Fergns Qian's user avatar
4 votes
1 answer
173 views

Convex hull of 3 points in Cartan-Hadamard manifolds

Can the convex hull of $3$ points in a Cartan-Hadamard manifold be smooth? A Cartan-Hadamard manifold $M$ is a complete simply connected manifold with nonpositive curvature (so it includes the ...
Mohammad Ghomi's user avatar
1 vote
1 answer
72 views

Is this notion of being "fully" convex closed under set addition?

While reading through "Linear Operators: General theory" by "Jacob T. Schwartz", reading the corollary to II.10.1 which states that for a compact convex subset $C$ of some ...
P. Quinton's user avatar
1 vote
1 answer
90 views

If $|P|<\infty$ and $C=P\cap\partial(\textrm{Conv}(P))$, then $P\subset\textrm{Conv}(C)$?

That is, if $P$ is a finite set, and $C$ is the set of points in $P$ which lie on the boundary of the convex hull of $P$, then is $P$ contained in the convex hull of $C$? It seems true intuitively. In ...
one-day-at-a-time's user avatar
2 votes
1 answer
166 views

Lipschitz smooth convex extension

Assume that convex $f: S \to \mathbb{R}$ with $L$-Lipschitz continuous gradient on some convex compact $S \subset \mathbb{R}^d$ is given. It would be very helpful if there existed function $F$ such ...
Dmitry Vilensky's user avatar
3 votes
1 answer
147 views

Is a compact set of extreme points contained in a compact face?

I have run into the following question in convex analysis, which I haven't found answered in the literature: Suppose that $K$ is a "nice-enough" non-compact convex subset of a Hausdorff ...
Sean's user avatar
  • 135
2 votes
1 answer
223 views

Boundary points in $\overline{\operatorname{conv}\{z_i\}_{i\in I}}$

Let $X$ be an infinitely-dimensional Banach space and $\{z_i\}_{i\in I}$ be a set of linearly independent points in $X_{\leq 1}$, the closed unit ball of $X$. $I$ the index set is not necessarily ...
Sanae Kochiya's user avatar
5 votes
1 answer
413 views

Is there a Borel measurable $f:\mathbb{R}^d \to \mathbb{R}^d$ such that $f(x) \in \partial \varphi (x)$ for all $x$?

Let $\varphi: \mathbb{R}^d \to \mathbb{R}$ be a convex function. The subdifferential of $f$ at $x$ is defined as $$ \partial \varphi (x) := \{z \in \mathbb{R}^d : \varphi(y) \geq \varphi(x) + \langle ...
Akira's user avatar
  • 1,179
0 votes
0 answers
55 views

Zero flux along lines

I am considering the $L^1$ ball in $\mathbb{R}^d$, and a conservative vector field $V$ on it, which arises as the gradient of a bounded, almost-everywhere Lipchitz-function. Denoting by $e_i$ as the i’...
Brendan Mallery's user avatar
2 votes
1 answer
146 views

Distance between convex hulls in a bounded closed convex set

Let $X$ be an infinite-dimensional Banach space and $C\subseteq X$ be a bounded closed convex subset. Let $\{z_i\}_{i\in\mathbb{N}}$ be a sequence of linearly independent points in $C$ and for each $n\...
Sanae Kochiya's user avatar
1 vote
0 answers
165 views

Monotone likelihood ratio of convolved power function kernel, $p\ge 3$

It was shown in a previous answer that for: $f(x)=|x|^p$, $\;x\in \mathbb{R}$, $\;p>2$, defining the density: $$p(\phi;\theta) = \Big(f\big(\hspace{-1pt}\cos(\phi-\theta)\big) - f\big( \hspace{-1pt}...
japalmer's user avatar
  • 321
3 votes
1 answer
37 views

A converse question about the polyhedrality under linear mapping

It is quite well known that for any polyhedral set $K$ and any linear mapping $A$, the set $AK$ is polyhedral and hence closed. I am more curious about the converse problem, namely: Suppose $K$ is a ...
Wenqing Ouyang's user avatar
1 vote
0 answers
53 views

Extension of averaged nonexpansiveness for mappings that are not self maps

Let $\mathcal{H}$ be a Hilbert space and let $\alpha \in (0,1)$. We say that an operator $f:\mathcal{H} \rightarrow \mathcal{H}$ is Nonexpansive if $\|f(x)-f(y)\|_{\mathcal{H}} \le \|x - y\|_{\...
mlbj's user avatar
  • 11
2 votes
1 answer
160 views

Log-concavity of the difference of the second anti-derivative of Gaussians

I would like to prove the following but I couldn't manage to do it. Let $a>b>0$ be two real numbers. Let $f$ be the function defined as: $$\forall \sigma>0, ~\forall x\in\mathbb{R},~f_\sigma(...
NancyBoy's user avatar
  • 393
0 votes
0 answers
133 views

Convex conjugate of the sum of smallest elements

I recently came across this problem to find the conjugate function of the sum of $r$ smallest elements in a vector $$f(x) = \sum_{i=n-r+1}^{n}x_{[i]} \text{ for } x \in \mathbb{R}^n$$ where $x_{[i]}$ ...
Ben Carter Jr.'s user avatar
1 vote
0 answers
266 views

Monotone likelihood ratio of a kernel based on $\log(\cosh(x))$

Let $f(x) = \log(\cosh(x))$, and define the kernel density: $$p_r(\phi;\theta) = \Big(f\big(r\cos(\phi-\theta)\big) - f\big(r\cos(\phi+\theta)\big) \Big)\hspace{0.5pt} \frac{\sin(2\phi)}{\sin(2\theta)}...
japalmer's user avatar
  • 321
1 vote
1 answer
131 views

Monotone likelihood ratio of densities based on power function

Given $p,\phi,\theta \in \mathbb{R}$ such that $p>2$ and $0 \le \phi,\theta\le \pi/2$ define the density function: $$f(\phi;\theta) = \mbox{$\Large\frac{1}{p B\big(\hspace{-1pt}\frac{3}{2},\frac{p+...
japalmer's user avatar
  • 321

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