Questions tagged [convex-analysis]

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24 votes
3 answers
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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
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
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
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
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
6 votes
3 answers
810 views

Mixtures of log-convex functions are log-convex: a reference

A referee of a submitted paper requested details on the statement that $\int_0^a e^{-tx^2}\,dx$ is log-convex in real $t$, for each $a>0$. While there are a number of ways to prove this statement, ...
Iosif Pinelis's user avatar
5 votes
2 answers
1k views

Minimum of squared sum minus sum of squares

I know that $$ \min_{\|x\|_2=1=\|y\|_2} \left(\sum_{k=1}^nx_ky_k\right)^2-\sum_{k=1}^nx_k^2y_k^2 \geq -1/2 $$ with equality whenever $|x_k|=\frac{1}{\sqrt{2}}=|y_k|$ for two coordinates. I'm ...
squattyroo's user avatar
5 votes
2 answers
1k views

Hausdorff dimension of convex set in ${\bf R}^n$

I want to know the smoothness of convex set in ${\bf R}^n$. Recall the following definition. Definition : $X$ is a bounded closed convex set in ${\bf R}^n$ if for $x$, $y\in X$, the any $d$-...
Hee Kwon Lee's user avatar
  • 1,060
4 votes
0 answers
181 views

improved regularization for $\lambda$-convex gradient flows

It is well-known that gradient-flows of convex functionals are "parabolic" in some vague sense, and accordingly solutions tend to regularize instataneously. In the abstract context of gradient flows ...
leo monsaingeon's user avatar
2 votes
0 answers
103 views

Optimization over a convex cone generated by a set is equal to optimization over the set

Within my research I found an important doubt and that prevents me from advancing, the context of my doubt is as follows: We considerer the following optimization problem $$ \left\{\begin{array}{cl} \...
matematicaActiva's user avatar
2 votes
0 answers
107 views

Maximization of an integral functional over a closed convex set

I want to maximize $$F(w):=\sum_{1\le i,\:j\le2}\int\lambda^{\otimes2}({\rm d}(x,y))\left(w_i(x)f_j(x,y)\wedge w_j(y)f_i(y,x)\right)g_{ij}(x,y)$$ over the closed convex set $$S:=\left\{w\in{\mathcal L^...
0xbadf00d's user avatar
  • 161
1 vote
1 answer
249 views

Can we invoke "almost supermartingale" Theorem for deterministic sequences?

Perhaps stupid question. Question: Can "almost supermartingale" theorem be equally applicable to prove the convergence of some algorithms solving non-random optimization problems? Attempt ...
user550103's user avatar
1 vote
1 answer
96 views

On the Lipschitz continuity of the unit-normal vector field of a polytope

Let $\mu$ be a probability measure on $\mathbb R^n$ and let $P$ be a compact polytope in $\mathbb R^n$. For any $x \in \mathbb R^n \setminus P$, let $p(x) \in P$ be (unique!) point in $P$ which is ...
dohmatob's user avatar
  • 6,706
1 vote
1 answer
172 views

Conditions for Lipschitzness of boundary normal vector, almost everywhere

Let $C$ be a nonempty closed subset of $\mathbb R^n$. It is known that any such set satisfies the following condition (Unique CPP a.e). For almost every $x \in \mathbb R^n$, there exists a unique ...
dohmatob's user avatar
  • 6,706
1 vote
1 answer
141 views

How can we calculate the generalized gradient of $L^2\ni x\mapsto a\min(x(s),by(t))$?

Let $(T,\mathcal T,\tau)$ be a measure space, $a,b\ge0$, $s,t\in T$ and $$f(x):=a\min(x(s),bx(t))\;\;\;\text{for }x\in L^2(\tau).$$ How can we calculate the generalized gradient $\partial_Cf(x)$ of ...
0xbadf00d's user avatar
  • 161
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
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
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
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
6 votes
2 answers
1k views

Quantitative version of Jensen's inequality?

I've been looking at situations where Jensen's inequality is almost tight, and found myself proving a lemma that I'm nearly certain exists somewhere in the literature. The specifics are as follows: ...
user29374's user avatar
  • 131
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
6 votes
2 answers
300 views

For most directions does the supporting hyperplane meeting a bounded convex set meet it in one point?

Let $C\subseteq \mathbb R^n$ be non-empty, convex and compact. For $v\in S^{n-1}$, let $H_v$ be the supporting hyperplane in the direction of $v$ (i.e., $H_v$ is the boundary of the smallest closed ...
Alexander Pruss's user avatar
5 votes
1 answer
2k views

When do convexity and lower semicontinuity imply continuity?

Let $X$ be a nonempty compact convex subset of a locally convex space. Say $X$ has the convex function property if every convex, lower semicontinuous $f:X\to\mathbb R$ is also continuous. Question: ...
e.lipnowski's user avatar
5 votes
2 answers
935 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
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
4 votes
2 answers
618 views

Existence of a strictly convex function interpolating given gradients and values

I'm wondering where to find a proof and reference for the following facts, which I feel sure must be true. (1) Suppose we are given a finite set of points in $\mathbb{R}^{d+1}$. For each point, we ...
usul's user avatar
  • 4,429
3 votes
3 answers
214 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
  • 839
3 votes
1 answer
360 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
  • 313
2 votes
2 answers
2k views

Hessian of function of covariance matrices

Suppose we have a typical logdet function $\mathcal{L}$ with respect to a covariance matrix $\mathbf{A}$, $$ \mathcal{L}(\mathbf{A}) = \log\vert \mathbf{I} + \mathbf{A}\mathbf{S} \vert - \mathbf{q}^T(\...
liubenyuan's user avatar
2 votes
1 answer
217 views

On the infimal convolution of two norms on $\mathbb R^n$

$\newcommand{\R}{\mathbb R}$For natural $n$, $a\in\R^n$, and real $t>0$, let \begin{equation*} K:=K_{n,t}(a):=\inf_{x\in\R^n}(\|a-x\|_2+t\|x\|_1), \end{equation*} \begin{equation*} M:=M_{n,...
Iosif Pinelis's user avatar
2 votes
0 answers
136 views

Covering a space by cones

Let $X\subset\mathbb{R}^n$ be some connected and bounded $n$-dimensional manifold, e.g. a space homeomorphic to an open/closed ball with possibly some parts of it being removed. I am interested in ...
user51223's user avatar
  • 3,071
2 votes
1 answer
1k views

Subgradient of Minimum Eigenvalue

Consider three $N \times N$ Hermitian matrices $A_0$, $A_1$, $A_2$. Consider the function \begin{align} f(t_1,t_2)=\lambda_{\text{min}}(A_0+t_1A_1+t_2A_2) \end{align} where $\lambda_{\text{min}}$ ...
dineshdileep's user avatar
  • 1,371
2 votes
0 answers
136 views

Convexity of volume in terms of a deformation

In the context of convex optimization and mechanics, I am interested in the convexity of the potential energy $U$ of a pressure acting over some volume $V$ enclosed by a surface. Here pressure can be ...
olek n's user avatar
  • 51
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
  • 433
1 vote
0 answers
117 views

Does the convex-hull of a set contain zero (II)?

Let $(\lambda_1, \cdots , \lambda_d) \vdash k$ be a partition of $k$ of length $d$. Suppose we have $k$ non-zero vectors $\alpha^{(j)} = (\alpha^{(j)}_1, \cdots , \alpha^{(j)}_d) \in \mathbb{Z}^d$, ...
SMD's user avatar
  • 480
1 vote
1 answer
54 views

Generalised Lebesgue transform continuous wrt. strict topology?

Let $X$ and $Y$ be $\sigma$-compact spaces, and $\mu$ [resp. $\nu$] be a regular Borel probability measure on $X$ [resp. $Y$]. For a bounded continuous $c:X\times Y\rightarrow\mathbb{R}$, consider the ...
fsp-b's user avatar
  • 421
1 vote
1 answer
171 views

Sufficient conditions for the boundary unit-normal vector field of a closed convex set to be Lipschitz continuous

This is followup to the following question: On the Lipschitz continuity of the unit-normal vector field of a polytope Let $C$ be a (nonempty) closed convex subset of $\mathbb R^n$. Note that to every ...
dohmatob's user avatar
  • 6,706
1 vote
1 answer
289 views

How to prove the convexity of a simple function involving a ratio of two polygamma functions?

Let \begin{equation*} \Gamma(z)=\int_0^{\infty}t^{z-1}\textrm{e}^{-t}\textrm{d}t, \quad \Re(z)>0 \end{equation*} and $$ \psi(z)=[\ln\Gamma(z)]'=\frac{\Gamma'(z)}{\Gamma(z)}. $$ In the literature, ...
qifeng618's user avatar
  • 828
1 vote
1 answer
107 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
  • 141
1 vote
0 answers
118 views

Generalization of concave envelope

Let $g:\mathbb R_+\to\mathbb R$ be a measurable function (which could be supposed to be bounded and Lipschitz if required). Let $\mathcal P$ be the collection of probability measures $\mu$ on $\mathbb ...
CodeGolf's user avatar
  • 1,837
1 vote
1 answer
189 views

Proof of extended version of non-random "almost supermartingale"

In this question, a non-random version of "almost supermartingale" theorem is proved. Here, I would like to extend/apply the non-random version to the slightly different situation. I wonder ...
user550103's user avatar
0 votes
1 answer
63 views

Continuity of generalised Legendre transform

Let $X$ and $Y$ be $\sigma$-compact spaces, and $\mu$ [resp. $\nu$] be a regular Borel probability measure on $X$ [resp. $Y$]. For a bounded continuous $c:X\times Y\rightarrow\mathbb{R}$, we consider ...
fsp-b's user avatar
  • 421
0 votes
1 answer
280 views

Exponential Convexity

$\textbf{Definition:}$ 1. A function $h : (a,b)\rightarrow\mathbb{R}$ is exponentially convex if it is continuous and $$\sum _{i, j=1}^n\xi_i\xi_jh(x_i+x_j)\geq 0,$$ for all $n\in\mathbb{N}$ and all ...
Shinning Star's user avatar
0 votes
4 answers
443 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
  • 2,169
0 votes
1 answer
78 views

Is the mapping $F(a):= \arg\min_{x \in \mathbb R^n} \|x-a\|_2 + \|x\|_1$ non-expansive?

Fix $a \in \mathbb R^n$ and let $\|\cdot\|$ be any norm on $\mathbb R$ (e.g $\ell_1$ norm). For any $a \in \mathbb R^n$, it is clear that the function $f_a(x) := \|x-a\|_2 + \|x\|$ is strictly convex ...
dohmatob's user avatar
  • 6,706