# Questions tagged [convex-analysis]

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517
questions

3
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0
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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,...

3
votes

0
answers

200
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+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 ...

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 ...

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{...

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 ...

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 ...

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 &...

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
$$\lim_{x\to+\infty} \big(f_k(x)-x\big)=0.$$
...

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
$$
\...

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 ...

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 = \...

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 ...

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 $x_1,x_2 \in \mathbb{R}$,...

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 ...

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 $\...

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 ...

0
votes

0
answers

79
views

### 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^...

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 ...

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 $\alpha, L >0$ ...

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 ...

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}...

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 \...

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)=...

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_{\...

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 \...

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,...

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
$$
\Pi_K(x) = \operatorname{argmin}_{k\in K} \|k-x\|
$$...

-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)>...

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\...

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 ...

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 $...

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
$$...

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(...

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)...

4
votes

1
answer

173
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### 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 ...

1
vote

1
answer

72
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### 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 ...

1
vote

1
answer

90
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### 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 ...

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 ...

3
votes

1
answer

147
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### 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 ...

2
votes

1
answer

223
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### 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 ...

5
votes

1
answer

413
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### 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 ...

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’...

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\...

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}...

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 ...

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\|_{\...

2
votes

1
answer

160
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### 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(...

0
votes

0
answers

133
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### 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 $x_{[i]}$ ...

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)}...

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+...