Questions tagged [convex-analysis]

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4 answers
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Elementary applications of Krein-Milman

This is a cross-post from MSE: Elementary applications of Krein-Milman. I'm starting to suspect that the question just doesn't really have a great answer, it's worth a try. Recall that the Krein-...
24 votes
3 answers
1k views

Average measure of intersection of a convex region with its translate

Let $\lambda$ denote the Lebesgue-measure on $\mathbb{R}^n$, and let $C\subset\mathbb{R}^n$ be a convex region. My question is about $$f(C):=\int_{C} \lambda(C \cap (x + C) ) \mathrm{d} x.$$ How ...
zref's user avatar
  • 343
23 votes
2 answers
2k views

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
21 votes
2 answers
1k views

A strange variant of the Gaussian log-Sobolev inequality

Let $\phi : \mathbb{R}^d \to \mathbb{R}$ be a convex function, and assume that it grows at most linearly at infinity for simplicity. Denote by $\gamma$ the standard Gaussian measure on $\mathbb{R}^d$, ...
Elwood's user avatar
  • 562
20 votes
3 answers
1k views

Convergence of convex functions

I can prove the following result. Theorem 1. Let $f_n:\mathbb{R}^n\to \mathbb{R}$ be a sequence of convex functions that converges almost everywhere to a function $f:\mathbb{R}^n\to\mathbb{R}$. Then ...
Piotr Hajlasz's user avatar
18 votes
1 answer
1k views

The Teichmüller space $T_g$ of a closed riemann surface $S_g$ of genus $g \geq 2$ can't be parametrized by $6g−6$ geodesic length functions

I asked this question almost a month ago on Math SE. After waiting three weeks for an answer or a comment, I opened a bounty on the question in hope that it might get an answer this way. The bounty ...
user331406's user avatar
18 votes
3 answers
915 views

Convex functions in convex sets

Suppose $\Omega \subset \mathbb{R}^n$ is some bounded, convex set. For which domains $\Omega$ is it true that for every convex function $f:\Omega \rightarrow \mathbb{R}$ the average of the function in ...
Stefan Steinerberger's user avatar
18 votes
1 answer
2k views

How bad can the second derivative of a convex function be?

One can easily construct an example of a measurable function $f:(a,b)\to \mathbb{R}$ which satisfies the following property: $$\label{p}\tag{P} f\notin L^1(I),\ \mbox{for each interval}\ I\subset (a,...
Tomás's user avatar
  • 409
17 votes
1 answer
832 views

Extreme points of convex compact sets

Preparing to a lecture on Krein--Milman theorem I read in W. Rudin's Functional analysis textbook (1973) that it is unknown whether any convex compact set in any topological vector space has an ...
Fedor Petrov's user avatar
15 votes
1 answer
878 views

Second order differentiability of convex functions

Let $f:\mathbb{R}^n\to\mathbb{R}$ be a convex function. Then $f$ is locally Lipschitz and hence differentiable a.e. (Rademacher). Let $E\subset\mathbb{R}^n$ be the set of points where $f$ is ...
Piotr Hajlasz's user avatar
14 votes
1 answer
4k views

Is KL divergence $D(P||Q)$ strongly convex over $P$ in infinite dimension

By KL divergence I mean $D(P||Q) = \int dP \log(\frac{dP}{dQ})$. I am looking for the conditions under which this strong convexity is true and possible references. I could not find an answer for ...
Maziar Sanjabi's user avatar
14 votes
0 answers
302 views

How large are the smallest-area projections of a high-dimensional convex body?

Let $B$ be a convex body in $\mathbb{R}^d$, equipped with its standard Euclidean form, and assume that $$\intop_B x \, dx = 0$$ $$\frac{1}{|B|_d} \intop_B x_i x_j \, dx = \delta_{ij},$$ a ...
Alexander Shamov's user avatar
13 votes
1 answer
1k views

Aleksandrov's proof of the second order differentiability of convex functions

Aleksandrov [A], proved a remarkable property of convex functions. Theorem. If $f:\mathbb{R}^n\to\mathbb{R}$ is convex, then for almost every $x\in\mathbb{R}^n$ there is $Df(x)\in\mathbb{R}^n$ and ...
Piotr Hajlasz's user avatar
13 votes
2 answers
649 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
13 votes
1 answer
1k views

Minimize sum of $\ell_2$ norm and linear combination, on simplex

Let $\Delta_n := \{x \in \mathbb{R}^n | x \ge 0, \sum_{1 \le i \le n}x_i = 1\}$ be the $n$-simplex. For $a, b \in \mathbb R^n$, with $\Delta_n \not \ni a$, consider the problem of computing the ...
dohmatob's user avatar
  • 6,706
13 votes
1 answer
2k views

An intuition for three different types of subgradients (proximal, regular, limiting)

I'm having a bit of difficulty getting my head around the different types of subgradients we're currently covering in a nonsmooth optimisation class I'm taking. These subgradients are (assume $x \in$ ...
Ben Stott's user avatar
  • 239
12 votes
2 answers
728 views

Geometric applications of Ekeland's variational principle

I'm looking for geometric applications of Ekeland's variational principle in order to see it at work in a context I'm familiar with. Let me recall the principle itself: Definition. Let $(X,d)$ be a ...
alvarezpaiva's user avatar
  • 13.2k
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 \\ \...
Pluviophile's user avatar
  • 1,404
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
10 votes
2 answers
892 views

Constructing an independent uniform random variable from two independent ones

Does there exist a continuous (differentiable) function $h:[0,1]\times [0,1] \to [0,1]$ such that if $\alpha,\beta\in [0,1]$ are independent and uniformly distributed on $[0,1]$, the random variable $...
Peter's user avatar
  • 355
10 votes
2 answers
2k views

Continuous functions with convex level sets

Assume that $f:\mathbb{R}^{2}\to \mathbb{R}$ is a continuous function such that each level set $f^{-1}(c)$ is a convex set. To what extent such functions are studied? In particular: Is there a ...
Ali Taghavi's user avatar
10 votes
3 answers
1k views

Efficient computation of "discrete infimal convolution"

This question arises from an application to graphical models in probability theory, but I have abstracted that part out so only algebra remains. Let $\mathbb{R}$ denote standard field of real numbers ...
Noah Stein's user avatar
  • 8,393
10 votes
1 answer
4k views

Uniform convergence of convex functions

It is a well-known result that if a sequence of convex function $f_n(\cdot)$ converges on a dense set $C'$ of an open set $C$, then the limit function $f$ exists on $C$, and the converge is uniform ...
Roy Han's user avatar
  • 589
9 votes
1 answer
912 views

Convexity of the product of two exponential matrices

Let $S\subset\mathbb{R}$ be a convex set and $\mathbb{S}^{n}$ be the set of real symmetric matrices of order $n\times n$. A matrix valued function $\Gamma: S \rightarrow \mathbb{S}^{n}$ is said to ...
Tadashi's user avatar
  • 1,580
9 votes
1 answer
712 views

property of convex functions

I am able to give a proof to the following inequality for convex functions. Most likely this is well known, but I am unable to find a reference. I would appreciate if someone more knowledgeable in the ...
Hammerhead's user avatar
  • 1,171
9 votes
2 answers
888 views

When is a mapping the proximity operator of some convex function?

Is there a characterization of mappings $p : \mathbb R^n \rightarrow \mathbb R^n$ which are proximity operators (in the sense of Moreau) of l.s.c (extended) real-valued functions ? That is, given $p : ...
dohmatob's user avatar
  • 6,706
9 votes
1 answer
456 views

In what sense is the Bayesian posterior mean a “convex combination”?

I asked this on math.stackexchange with no response, I'm hoping someone here might have something. Suppose I want to estimate $x \in \mathbb{R}^n$ from two signals with zero mean, normally ...
Ronaldo Carpio's user avatar
9 votes
0 answers
135 views

A self-isometry of the sphere of a strictly convex Banach space that does not move basic vectors

Problem. Let $n\in\mathbb N$, $X$ be a strictly convex $n$-dimensional real Banach space, $S_X=\{x\in X:\|x\|=1\}$ be the unit sphere of $X$, and $e_1,\dots,e_n\in S_X$ be linearly independent points. ...
Taras Banakh's user avatar
  • 40.7k
9 votes
0 answers
910 views

Strong convexity of the trace of the square root of a matrix function

Any clues about how to prove that the following function is strongly-concave in $x$? (We conjecture it is $2$-strongly concave but cannot prove it. We have already proved strict concavity through ...
Mary's user avatar
  • 91
8 votes
1 answer
658 views

Is the square root of the Kullback-Leibler divergence a convex map?

$\newcommand{\KL}{\operatorname{KL}}$Let $X$ be a Polish metric space and $P(X)$ the space of probability measures on $X$. Given $\mu, \nu\in P(X)$, recall that $$\KL(\mu\parallel\nu) = \begin{cases}\...
ECL's user avatar
  • 271
8 votes
1 answer
478 views

Concavity of the trace of a matrix power

Let $B$ be an $n\times n$ matrix, and define $f$ to be the function that maps positive semidefinite (PSD) $n\times n$ matrices $A$ to real numbers by $$ f(A) = \mathrm{trace}( (B^*A^2B)^{1/3}). $$ ...
Sasho Nikolov's user avatar
8 votes
2 answers
696 views

On the convexity of element-wise norm 1 of the inverse

Question first asked on math.stackexchange here: https://math.stackexchange.com/questions/317209/on-the-convexity-of-element-wise-norm-1-of-the-inverse On the convexity of element-wise norm 1 of the ...
Ferpect's user avatar
  • 83
8 votes
1 answer
578 views

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
  • 183
8 votes
1 answer
1k views

Uniqueness of a Solution for a Convex Optimization Problem

I have the following convex optimization problem: $$\begin{array}{ll} \text{maximize}_{{f,g}} & \displaystyle\int_{\Omega} g^u{f}^{1-u}\mathrm{d}\mu\\ \text{subject to} & \displaystyle\int_{\...
Seyhmus Güngören's user avatar
8 votes
0 answers
192 views

Concavity of product and ratio of sums

Apologies if this question is not appropriate for MathOverflow. I have asked at Math.StackExchange without success. Consider the function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ defined as $$ f(x)=\...
user_lambda's user avatar
7 votes
2 answers
544 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
  • 6,611
7 votes
1 answer
522 views

Asking for an English version of Aleksandrov's famous 1939 paper in Convex Geometry

I have difficulty even in finding a Russian version of the next paper: "Aleksandrov, A. D., Almost everywhere existence of the second differential of a convex function and some properties of ...
Wenqing Ouyang's user avatar
7 votes
1 answer
2k views

Does midpoint-convex imply rationally convex?

Say that a function $f$ defined on $\mathbb{Q}^n$ is midpoint convex if $f((x+y)/2) \le (f(x) + f(y))/2$. Say that it is rationally convex if, for $\lambda \in \mathbb{Q} \cap [0,1]$ and $\bar \lambda ...
Dylan Thurston's user avatar
7 votes
1 answer
782 views

Compactness of set of indicator functions

Let $\chi_A(x)$ denote an indicator function on $A\subset [0,1]$. Consider the set $$K=\{\chi_A(x): \text{ A is Lebesgue measurable in }[0,1]\}.$$ Is this set compact in $L^\infty(0,1)$ with respect ...
Saj_Eda's user avatar
  • 395
7 votes
3 answers
598 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
7 votes
2 answers
438 views

Proving the set $\left\lbrace \frac{(x + y)^2}{\sqrt{y}} \leq x - y + 5, y > 0 \right\rbrace$ is convex

I have recently picked up a course on Convex Analysis in my spare time, but feel I'm not quite up to speed with the 'tricks' for proving a set is convex. I have managed to prove this by moving all ...
AlwaysLost123's user avatar
7 votes
1 answer
374 views

Nondifferentiable convex function whose subdifferential admits a continuous selection

Is there a convex function $F$ that is not differentiable, but whose subdifferential admits a continuous selection, i.e. a continuous $g$ with $g(x) \in \partial F(x)$ for all $x$ in the domain? In ...
usul's user avatar
  • 4,429
7 votes
2 answers
552 views

Gaussian and the convex hull of moment curves

Let $c_1,\dots, c_d$ be the first $d$ moments of the standard normal distribution. Does the point $(c_1,\dots, c_d)$ lie in the convex hull of the set $\{(t,t^2,\dots,t^d)\colon t\in[-b,b]\}$, for a ...
Minkov's user avatar
  • 1,117
7 votes
1 answer
184 views

Two (new?) variants of convex functions

I find that the following two types of functions are useful to my research. (i) We know that a function $f: \mathbb{R}_+^m\rightarrow \mathbb{R}$ is called convex if for all ${\bf x,y}\in \mathbb{R}...
Zhigang Cao's user avatar
7 votes
1 answer
278 views

A question on regularity of the Legendre transform

Let $f(x)$ be a strictly convex real-valued $C^{\infty}$ function on an open neighborhood of the origin in $\mathbb R^n$ with $f (0, \ldots , 0)= \partial_j f(0, \ldots , 0)=0$ for all $j$. If the ...
Jan Boman's user avatar
  • 585
7 votes
0 answers
400 views

Are there any characterizations of $C^2$ convex functions?

There are several characterizations of convex functions with the Lipschitz continuous gradient. If we already know that the function is of class $C^1$, then we have the following equivalent conditions:...
Piotr Hajlasz's user avatar
7 votes
0 answers
869 views

Geometry of level sets of a convex function

EDIT: Let $\Omega\subset \mathbb{R}^n$ be an open subset. Let $f\colon \Omega\to\mathbb{R}$ be a function such that for some $\lambda$ the function $f(x)+\lambda |x|^2$ is convex. Assume that the ...
asv's user avatar
  • 21.1k
7 votes
0 answers
228 views

orthogonal projector onto the set of convex functions

Let $\Omega\subset \mathbb R^d$ be an open, convex domain, and consider the Hilbert space $L^2(\Omega)$. Each sum of convex functions is convex, hence the subset $Conv(\Omega)$ of all convex functions ...
Delio Mugnolo's user avatar
6 votes
2 answers
532 views

Conditions for including cones

Consider $N$ $n$-dimensional vectors, where the angle between any two vectors is acute and their starting point is at the origin. Can we rotate these vectors together so that the coordinate components ...
dzk's user avatar
  • 61
6 votes
2 answers
512 views

Why are $\Gamma_0$ functions called this

It is very common to indicate with $\Gamma_0(A)$ the set of lower semicontinuous convex functions from $A$ to $(-\infty,+\infty]$ with nonempty domain. An example of usage of this notation can be ...
MMFF's user avatar
  • 71

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