Questions tagged [convex-analysis]
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45
questions
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 ...
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 ...
15
votes
1
answer
913
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 ...
13
votes
1
answer
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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 ...
7
votes
0
answers
404
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:...
7
votes
3
answers
846
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, ...
5
votes
2
answers
1k
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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 $d$-...
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 ...
4
votes
0
answers
184
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 ...
2
votes
0
answers
109
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 $$S:=\left\{w\in{\mathcal L^...
2
votes
0
answers
103
views
Optimization over a convex cone generated by a set is equal to optimization over the set
Within my research I found an important doubt and that prevents me from advancing, the context of my doubt is as follows:
We considerer the following optimization problem
$$
\left\{\begin{array}{cl} \...
1
vote
1
answer
252
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 ...
1
vote
1
answer
100
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 ...
1
vote
1
answer
178
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 ...
1
vote
1
answer
142
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 ...
14
votes
0
answers
304
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 ...
9
votes
1
answer
718
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 ...
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 ...
7
votes
2
answers
560
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 ...
6
votes
2
answers
303
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 ...
6
votes
2
answers
1k
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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: ...
6
votes
2
answers
519
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 ...
5
votes
2
answers
963
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\...
5
votes
1
answer
364
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 ...
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: ...
4
votes
2
answers
634
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 ...
3
votes
3
answers
219
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 ...
3
votes
1
answer
363
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 ...
2
votes
1
answer
220
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,...
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 $\mathbf{A}$,
$$
\mathcal{L}(\mathbf{A}) = \log\vert \mathbf{I} + \mathbf{A}\mathbf{S} \vert - \mathbf{q}^T(\...
2
votes
0
answers
136
views
Covering a space by cones
Let $X\subset\mathbb{R}^n$ be some connected and bounded $n$-dimensional manifold, e.g. a space homeomorphic to an open/closed ball with possibly some parts of it being removed.
I am interested in ...
2
votes
0
answers
136
views
Convexity of volume in terms of a deformation
In the context of convex optimization and mechanics, I am interested in the convexity of the potential energy $U$ of a pressure acting over some volume $V$ enclosed by a surface. Here pressure can be ...
2
votes
1
answer
1k
views
Subgradient of Minimum Eigenvalue
Consider three $N \times N$ Hermitian matrices $A_0$, $A_1$, $A_2$. Consider the function
\begin{align}
f(t_1,t_2)=\lambda_{\text{min}}(A_0+t_1A_1+t_2A_2)
\end{align}
where $\lambda_{\text{min}}$ ...
1
vote
1
answer
56
views
Generalised Lebesgue transform continuous wrt. strict topology?
Let $X$ and $Y$ be $\sigma$-compact spaces, and $\mu$ [resp. $\nu$] be a regular Borel probability measure on $X$ [resp. $Y$].
For a bounded continuous $c:X\times Y\rightarrow\mathbb{R}$, consider the ...
1
vote
1
answer
111
views
Monotone likelihood ratio of densities based on power function
Given $p,\phi,\theta \in \mathbb{R}$ such that $p>2$ and $0 \le \phi,\theta\le \pi/2$ define the density function:
$$f(\phi;\theta) =
\mbox{$\Large\frac{1}{p B\big(\hspace{-1pt}\frac{3}{2},\frac{p+...
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 $\alpha^{(j)} = (\alpha^{(j)}_1, \cdots , \alpha^{(j)}_d) \in \mathbb{Z}^d$, ...
1
vote
1
answer
296
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, ...
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 ...
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 ...
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 ...
1
vote
1
answer
175
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 ...
0
votes
1
answer
281
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 ...
0
votes
1
answer
79
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 ...
0
votes
4
answers
444
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 ...
0
votes
1
answer
66
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 ...