Questions tagged [convex-analysis]
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563 questions
1 vote
0 answers
71 views
Vanishing of Monge–Ampère operator on a convex function
Let $\Omega_1,\Omega_2\subset\mathbb{R}^n$ be strictly convex bounded domains, possibly with smooth boundary, and such that $\bar\Omega_1\subset\Omega_2$. Let $u\colon \bar\Omega_1\to \mathbb{R}$ be a ...
- 23k
2 votes
3 answers
118 views
Norm of metric projections along rays
Consider a non-empty, closed, convex set $C\subseteq\mathbb R^d$ ($d\geq 1$) containing the origin and let $\pi$ be the metric projection onto $C$. Fix some $x\in\mathbb R^d$. Let $f(t)=\|\pi(tx)\|_2$ ...
- 225
4 votes
1 answer
404 views
A general result on the convex function
During my research, I came a cross this question : Let $f \in C([0,1])$ convex. Is it true that $$\int_0^1 \max(f(t),f(1-t)) \geq\\ 2/3\max(f(1/3),f(2/3))+1/3 \max(f(1/6),f(5/6))?$$
- 5,881
3 votes
2 answers
270 views
Convex sets, tangent cones and convergence
Let $K\subseteq\mathbb R^d$ be a non-empty, closed, convex set and $x_0\in \partial K$ (the boundary of $K$). Let $u\in\mathbb R^d$ be a vector satisfying $\langle u,x-x_0\rangle\leq 0$ for all $x\in ...
- 225
3 votes
0 answers
144 views
Countable requirements for the boundary of a convex set to be smooth
Consider a random compact convex set $H\subset\mathbb{R}^d$. I can show that, for any given direction $\theta\in\mathbb{S}^{d-1}$, with probability 1, the $\theta$-exposed face (the set of points $x$ ...
1 vote
1 answer
273 views
Is there anything like expectation for set-valued random variables?
Suppose $(\Omega, \mathcal{F}, \mathbb{P})$ is a probability space. Let $f:\mathbb{R}^n \to \mathbb{R}$ be a real-valued function, $\varphi:\mathbb{R}^n \to \mathsf{P}(\mathbb{R}^n)$ be a set-valued ...
- 339
3 votes
1 answer
286 views
$C^2-$regularity of this function
Let $g:E\to\mathbb R_+$ be a probability density supported on $E$, i.e., $$\int_E g(x)dx=1,\quad \int_E |x|^2 g(x)dx<\infty,$$ where $E\subset\mathbb R^d=:\Omega$ is connected and open. Define $F:\...
- 1,416
3 votes
0 answers
108 views
Duality for "sum of squares" tensors
Let $T$ be a symmetric tensor in $\left( \mathbb{R}^d \right)^{\otimes n}$ such that the polynomial $$\sum_{i_1,i_2, \ldots i_n}T_{i_1i_2 \ldots i_n} x_{i_1}^2 x_{i_2}^2 \ldots x_{i_n}^2$$ is a sum ...
1 vote
0 answers
70 views
Covering the space with shifts of $sK'$ where $K'$ is the polar body of $K$ and $s>0$: how large does $s$ need to be?
Let $K \subset \mathbb R^n$ be a symmetric convex body with the following property: if $a,b \in \mathbb Z^n$ are distinct, then $(K+a) \cap (K+b)$ intersect in a set of zero Lebesgue measure. In other ...
- 137
3 votes
1 answer
153 views
Exchanging norms of proximal operators
I have two friendly functions $g,h:\mathcal{X}\subseteq\mathbb{R}^N\to\mathbb{R}$ whose exact properties I'm somewhat flexible on. Maybe for starters they are lower semi-continuous and convex. Their ...
0 votes
0 answers
74 views
When is $f(x) - g(x)$ strongly convex? ($f$ exponential, $g$ Beta/Gamma-based)
Let $r \in (-1,1)$ be fixed. Consider the two functions: $$ f(x) := (1 + e^x + e^{-x}) \ln 2 - e^x \ln(1 + r) - e^{-x} \ln(1 - r), $$ and $$ g(x) := -\ln B(1 + e^x,\ 1 + e^{-x}) = -\ln \Gamma(\alpha(x)...
- 2,602
3 votes
0 answers
225 views
Measurable selection of subdifferential of convex function
Let $(X, \Sigma)$ be a measurable space. $ f: X \times \mathbb{R}^{n} \rightarrow \mathbb{R}$ is a $\Sigma \otimes \mathcal{B}\left(\mathbb{R}^{n}\right)$-measurable function and $\forall x \in X, y \...
- 93
2 votes
1 answer
157 views
About extreme of the duality mapping in $C_0(\Omega)$
Let $\Omega$ be locally compact topological space. Consider $C_0(\Omega)$ being the space of continuous functions $u$ on $\Omega$ that vanish at infinity, that is, for $r>0$ there is a compacte set ...
- 1,155
2 votes
0 answers
195 views
Extreme points of the set of **simple fusions**
We define the simple fusion of a finite-supported measure as follows (similar to Elton and Hills 1998): Suppose $P$ and $Q$ are probability measures in $\mathbb{R}^d$ with finite supports $\mathrm{...
- 21
4 votes
2 answers
191 views
Subdifferential $\partial \varphi(u)$ admits a measurable selection, where $\varphi:\mathbb{R}^3\to \mathbb{R}$ is convex and $u$ is measurable
I am looking for a reference for the following claim: Let $\Omega\subset \mathbb{R}^3$ be a bounded domain, $u:\Omega\to \mathbb{R}^3$ be measurable and let $\varphi:\mathbb{R}^3 \to \mathbb{R}$ be ...
- 135
12 votes
1 answer
411 views
Is there a face of the unit ball in a reflexive Banach space without extreme points
Let $X$ be a Banach space with unit ball $B_X:=\{x \in X \colon \|x\| \le 1\}$. A face $F \subset B_X$ is a non-empty convex subset such that if $tx + (1-t)y \in F$ for some $x,y \in B_X$ and any $t \...
- 857
1 vote
0 answers
62 views
Vanishing curvature of the lower convex envelope when it differs from the original function
I've been told to split up a previous post in order to maintain the principle that $1\text{ post} = 1 \text{ question}$. This question is therefore a resubmission of Property 2 from the previous post; ...
1 vote
1 answer
97 views
Properties of the set on which the lower convex envelope differs from the original function
Motivation In thermodynamics, given a free-energy function $f$ of a conserved parameter $\phi$ for some material, we can predict whether that material will prefer to separate into multiple phases from ...
0 votes
1 answer
178 views
Characterization of convex positively homogeneous functions according to behavior on unit sphere
For now just speaking in $\mathbb{R}^2$ we call a function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ positively homogeneous if for every $\lambda \ge 0$ we have $f(\lambda x )=\lambda f(x)$. For such a ...
- 45
2 votes
0 answers
98 views
Extreme points of the unit ball of the James space $J$ and the James tree space $JT$
Consider the $J$ sequence space and $JT$ the James tree space. Both are separable non-reflexive Banach spaces containing no isomorphic copies of $\ell_1$. Moreover, $JT$'s dual is non-separable, ...
- 2,602
3 votes
1 answer
247 views
Convexity of rational function?
I am studying the following function but I have trouble checking if it is convex or not... For $N>1$, we let $A$ be a $N$ by $N$ matrix with positive coefficients. For $X \in \mathbb{R}^N$ with ...
- 157
4 votes
1 answer
156 views
What is the closed convex hull of convex ridge functions?
Crossposted at Mathematics SE Let us define the following class of functions: $$ \mathcal{F} := \left\{ f : \mathbb{R}^d \to \mathbb{R} \,\middle|\, f(x) = \sum_{i=1}^n \varphi_i(\xi_i^\top x),\ \...
- 41
7 votes
1 answer
520 views
A particular continuous selection problem
Let $B$ and $S:=\partial B$ denote, respectively, the unit ball and the unit sphere wrt to some norm $\|\cdot\|$ on $\Bbb R^2$. For a fixed real $r\in(0,1]$ and all $u\in S$, let $$D_u:=(B+ru)\cap(B-...
- 142k
5 votes
1 answer
300 views
A continuous selection problem
Let $(B_t)_{t\ge0}$ be a continuous (say wrt the Hausdorff metric) family of nonempty centrally symmetric convex compact subsets of $\Bbb R^2$. For a norm $\|\cdot\|$ on $\Bbb R^2$ and each real $t\...
- 142k
0 votes
1 answer
85 views
Convex conjugate of sum of functions of overlapping pairs of variables
Let $ f: (0, \infty)^{2} \mapsto (0, \infty) $ smooth, non-negative, convex. Define $F(x_1, x_2, x_3):(0, \infty)^{3} \rightarrow [0, \infty) = f(x_1, x_2) + f(x_1, x_3) + f(x_2, x_3)$. How to express ...
- 119
1 vote
0 answers
53 views
Extension of the Ioffe-Tikhomirov theorem to locally Lipschitz functions with the Clarke subdifferential?
More precisely, assume that $f:X\times Y\to \mathbb{R} \cup \{+\infty \}$ is a function where $Y\subset X$ is compact, $\partial_{x}^{C}f\left( x_{0},y\right) $ is the Clarke subdifferential of at $...
0 votes
1 answer
213 views
What are the real numbers $a,b,c,f,g,h$ that make the convex hull generated by the $4$ common real points of $g_1=0$ and $g_2=0$ contain $(0,0)$?
In a previous question, I asked whether $(0,0)$ need to be in the convex hull generated the by common points of the quadratic curves $$ \begin{align} g_1&:=x y-a-b x-c y+a x^2=0 \\ g_2&:=y^2-...
1 vote
2 answers
236 views
Is $(0,0)$ within the convex hull generated by the real variety generated by the quadratics $g_1$ and $g_2$?
I have the following polynomials in $\mathbb{R}[x,y]$ $$ \begin{aligned} g_1 &= xy - a - b x - c y +a x^2 \\ g_2 &= y^2 - h - f x - g y + (h-1) x^2 \end{aligned} $$ where $a,b,c,f,g,h \in \...
2 votes
0 answers
114 views
A special convex cone of convex functions
This question is related to On the convex cone of convex functions Let $\Omega \subset \mathbb R^d$ be a closed set. Define $\mathcal F$ to be the set of (continuous) convex functions on $\Omega$, and ...
- 1,693
3 votes
0 answers
274 views
Show that $v\mapsto\mathbb{E} \operatorname{Prox}_{v h}^{\prime}\left(Y+v\cdot \sigma \mathrm{Z}\right)$ is a decreasing function
Show that for positive convex function $h$, $$v\mapsto\mathbb{E} \operatorname{Prox}_{v h}^{\prime}\left(Y+v\cdot \sigma \mathrm{Z}\right)$$ is a decreasing function on $v\in (0,+\infty)$ where $\...
- 57
2 votes
2 answers
172 views
Gradient norm of convex function restricted to points bounded away from argmin is minimized at the boundary of the restriction
Say $f: \mathbb{R}^{d} \to \mathbb{R}$ is convex and differentiable everywhere. Suppose also that $f$ is minimizable, and denote $argmin f := S$ (we may assume $S$ is bounded). Pick any $\delta >0$....
- 21
0 votes
0 answers
74 views
Some kind of scaled convexity
I have the following discrepancy that satisfies this type of inequality. I want to know if this can be related to some kind of weak convexity. If so, what is the name of this property? Additionally, ...
- 105
-4 votes
1 answer
255 views
Inequalitie of Jensen 2.0
During my research I came across this question. A) $(h,g) \in C^2([0,1], \mathbb R_+)^2$ with $\forall x \in [0,1], 2(h'(x))^2 \leq h''(x) \times h(x)$ and $2(g'(x))^2 \leq g''(x) \times g(x)$ Is it ...
- 5,881
0 votes
0 answers
47 views
Maximal property of c-cyclically monotone sets
Let $X$ and $Y$ be finite sets and $c:X\times Y\rightarrow \mathbb{R}$ be a real cost function. A subset $\Gamma\subseteq X\times Y$ is $c$-cyclically monotone if for all $n$ and $\{(x_i,y_i)\}_{i=1}^...
- 1
1 vote
1 answer
259 views
Second directional derivative of convex conjugate
Let $f: [0, 1] \to \mathbb{R}$ be continuous. Define the conjugate $$f^*(a) := \sup_{t \in [0, 1]} \{at - f(t)\}$$ and so $f^*$ is closed convex (even though $f$ is nonconvex). I would like to figure ...
- 95
2 votes
1 answer
182 views
Reference request for elementary convex geometry property
I need to use the following lemma for a proof. It is an elementary result, which I am sure is well known, just I am not familiar enough with the relative literature to find a direct reference. Is ...
- 345
2 votes
1 answer
154 views
Does approximately null gradient imply approximately global minimum for convex functions?
Let $f: \mathbb{R}^{n} \rightarrow \mathbb{R}_{+}$ be a non-negative and differentiable convex function which vanishes in a non-empty convex set $\Omega$ - possibly unbounded. Usually, when one ...
- 225
2 votes
0 answers
99 views
Two-terms Euler-Maclaurin formula for concave functions over polytopes
Let $P\subset\mathbb{R}^{n}$ be a lattice polytope (vertices are in $\mathbb{Z}^{n}$). Set $P_{k}=P\cap k^{-1}\mathbb{Z}^{n}$ for $k\geq1$. Given a concave function $\phi:P\rightarrow\mathbb{R}$ (...
- 63
1 vote
0 answers
90 views
Behaviour of the solutions of parametrized multivariable non-linear (non polynomial) system of equations
The following problem arose out of a research problem. Let us consider the $n \times n$ matrix valued function $[x_{i,j}(p)]$ (of $p$), satisfying $$ \sum_j x_{i,j}(p) x_{k,j}(p)|x_{k,j}(p)|^{p}= \...
- 755
1 vote
1 answer
93 views
Envelopes of functions with respect to some convex cone $\mathcal{F}$
Let's say we are given a function $f:\mathbb R ^d\to \mathbb R$ continuous. Assume that $\mathcal F$ is a convex cone of continuous functions ($\mathbb R^d$ to $\mathbb R$) closed under maxima. I am ...
- 301
2 votes
1 answer
171 views
Points of differentiability of convex functions
Let $U$ be an open neighbourhood of $0 \in \mathbb{R}^2$ and $f\colon U \to \mathbb{R}$ a convex (and bounded) function. Denote by $D \subset U$ the set of points on which $f$ is totally ...
- 3,128
7 votes
0 answers
314 views
Proving this function is convex
Let $C$ be a symmetric positive definite matrix such that $0\leq c_{ij} \leq 1$, $c_{ii}=1$, and define $f$ as $$f(x)=\sum_{i}x_{i}\log(\sum_{j}c_{ij}x_{j})$$ for positive vectors $x$ (in fact let's ...
- 4,131
12 votes
2 answers
554 views
On the convex cone of convex functions
$\newcommand\R{\Bbb R}$Let $F$ be the set of all functions of the form $\max(a,b,c)$, where $a,b,c$ are affine functions from $\R^2$ to $\R$ and the maximum is taken pointwise. Let $G$ be the set of ...
- 142k
2 votes
0 answers
101 views
Test probability distributions increasing in convex order on $\mathbb R^2$?
Two probability distributions $\mu, \nu$ on $\mathbb R^d$ are said to be increasing in convex order if $$\int_{\mathbb R^d} |x|\mu(dx) + \int_{\mathbb R^d} |x|\nu(dx)<\infty$$ and $$\int_{\mathbb R^...
- 1,416
1 vote
0 answers
65 views
Convex 1-semiconcave functions : extreme points
Let $C$ be the set of functions $f:\mathbb{R}^n \to \mathbb{R}$ such that $x\mapsto f(x)$ and $x\mapsto \lVert x \rVert^2 - f(x)$ are both convex. If $f$ belongs to $C$, then $f$ plus any affine ...
- 2,927
1 vote
0 answers
159 views
Convexity and subdifferential monotonicity
Do you know any reference where I can find some results in this sense: Consider $W:K\to [0,\infty)$ is a functional defined on a convex cone $K\subset X$, where $X$ is a Banach space. Then the ...
- 2,029
19 votes
2 answers
2k views
Probability vector $p$ majorizes its normalized entropy vector $\small \frac{-p\log p}{H(p)}$
I guess the following inequality $$ \sum_{i=1}^n g \left (\frac{-p_i \log p_i}{H(\boldsymbol{p})} \right ) \le \sum_{i=1}^n g (p_i)$$ holds for any continuous convex function $g$ and any probability ...
- 413
6 votes
0 answers
372 views
Distribution class closed under convolution counterexample?
Define the class of probability density functions $\mathcal{C}$: $\,p \in \mathcal{C}$ iff $p(x)=p(|x|)$, and $\log p(\!\sqrt{x})$ is convex on $[0,\infty)$. Conjecture: if $p,q \in \mathcal{C}$, then ...
- 401
2 votes
0 answers
83 views
An s-convex function lying between two convex functions
Let $f: \mathbb R_{+} \to \mathbb R_{+}$ be an $s$-function in the second sense, i.e., $$ f(\lambda x +(1-\lambda)y) \leq \lambda^s f(x) +(1-\lambda)^s f(y)$$ for every $\lambda \in (0,1)$. Assume ...
- 55
2 votes
0 answers
175 views
Continuity of entropies, replica trick and Hausdorff moment problem
I could not find a really appropriate title for my question (happy to revise) but let me explain. Suppose $p(x|c)$ is a probability density function over $x \in [0,1]$ which depends continuously on ...
- 231