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28 votes

Are such functions differentiable?

For any $x$ and for sufficiently large $n$ such that $1+x/n>0$, it holds that \begin{align} f(x) &\ge f\left (\frac{(n-1)x}n \right) (1+x/n)\\ &\ge f\left (\frac{(n-2)x}n \right)(1+x/n)^2 \\...
Hhan's user avatar
  • 561
23 votes
Accepted

Average measure of intersection of a convex region with its translate

The following proposition answers OP's question regarding the upper bound of $$\tau(C) \Doteq f(C)/\lambda^2(C).$$ Let $B_n$ be the closed Euclidean unit ball of $\mathbb{R^n}$ centred at $0$, that is ...
Luc Guyot's user avatar
  • 7,578
22 votes
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Extreme points of convex compact sets

A counterexample is given in the following paper: Roberts, James W. "A compact convex set with no extreme points." Studia Mathematica 60.3 (1977): 255-266. I did not see the original paper, but ...
Michael Greinecker's user avatar
19 votes
Accepted

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 think Scott's argument is that the lengths of $6g-6$ curves can't form coordinates for Teichmuller space. If one has $6g-6$ geodesics which parameterize, then they must be filling (they meet every ...
Ian Agol's user avatar
  • 67.4k
16 votes
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How bad can the second derivative of a convex function be?

The second derivative of a convex function, in the distributional sense, is a non-negative bounded measure. And conversely. If this measure $\mu$ contains a sum $\sum_na_n\delta_{x=x_n}$, where $(x_n)...
Denis Serre's user avatar
  • 51.7k
15 votes
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Convex functions in convex sets

Zachary Chase said that he is still interested, so I'll make a post. It is a bit on the long side and I'll need to draw a few pictures to make it comprehensible, so today I'll just post a ...
fedja's user avatar
  • 60.3k
13 votes
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Asking for an English version of Aleksandrov's famous 1939 paper in Convex Geometry

This paper is contained in the 1-st volume of Aleksandrov's Selected works. This has an English translation: Alexandrov, A. D. Reshetnyak, Yu. G. (ed.) Selected works. Part 1: Selected scientific ...
Alexandre Eremenko's user avatar
12 votes

Compactness of set of indicator functions

This set is not closed in the weak* topology. Indeed, for each $n$ consider the set $A_n=: [0, \frac{1}{2n})\cup [\frac{2}{2n}, \frac{3}{2n}] \cup\dots\cup [\frac{2n-2}{2n}, \frac{2n-1}{2n})$, i.e. ...
Mateusz Wasilewski's user avatar
12 votes
Accepted

Does midpoint-convex imply rationally convex?

Assume that $g$ defined on $\mathbb{Q}^n$ is midpoint convex. First we show that we can extend the midpoint inequality to arbitrary means: $g((x_1+\dots+x_m)/m)\leq (g(x_1)+\dots+g(x_m))/m$ for any $...
Ivan Meir's user avatar
  • 4,802
12 votes
Accepted

Is a function of several variables convex near a local minimum when the derivatives are non-degenerate?

Let $$\begin{aligned} f(x,y) & = x^4 - x^2 y^2 + y^4 \\ & = \tfrac{1}{2} x^4 + \tfrac{1}{2} y^4 + \tfrac{1}{2} (x^2 - y^2)^2 . \end{aligned}$$ Then $f$ is a strictly positive (except at the ...
Mateusz Kwaśnicki's user avatar
12 votes
Accepted

Convexity and Lipschitz continuity

That's a standard result in convex optimization. For example Theorem 2.1.5 in Nesterov's "Introductory Lectures on Convex Optimization" states that the following are equivalent: $f$ is $C^1$...
Dirk's user avatar
  • 12.4k
12 votes

Conditions for including cones

Iosif Pinelis already gave a nice solution to show that the answer is "no" for sets of infinitely many vectors, and thus for $N$ very large. I'll show that the answer is also "no" ...
Nathaniel Johnston's user avatar
11 votes
Accepted

Why are $\Gamma_0$ functions called this

I think Carlo Beenakker digged up the right reference for the notation of the set, but I think more can be said. First, there is some meaning for the subscript $0$ which can be found in the same ...
Dirk's user avatar
  • 12.4k
11 votes
Accepted

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

The paper On the second differentiability of convex surfaces by Bianchi, Colesanti, and Pucci (Geometriae Dedicata volume 60, pages 39–48 (1996)) concerns the proof of the Busemann-Feller-Alexandroff ...
Willie Wong's user avatar
10 votes
Accepted

Concavity of the trace of a matrix power

Unfortunately, the conjectured function is not concave. Here is a simple simpler counterexample. \begin{equation*} B = \begin{bmatrix} 1 & 2 \\ 3 & 4\end{bmatrix},\quad A = \begin{bmatrix} ...
Suvrit's user avatar
  • 28.4k
10 votes
Accepted

When minimum of two supporting functionals of convex bodies is convex?

Yes. At first, if $h=\min(h_A,h_B)$ is convex (note that it is also 1-homogeneous), it is a support function of the body $C:=\{x:\forall\xi\in \mathbb{R}^n,\langle \xi,x\rangle\leqslant h(\xi)\}$. ...
Fedor Petrov's user avatar
10 votes
Accepted

property of convex functions

Anyway, If you know a 5 line proof for the first inequality please share it with us OK, here goes. Assume that $\inf_P f=-1$ and that it is (nearly) attained at the origin. Let $K=\{x\in P: f(x)\le ...
fedja's user avatar
  • 60.3k
10 votes
Accepted

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

$\newcommand\de\delta\newcommand{\KL}{\operatorname{KL}}\newcommand{\p}{\,\|\,}$The maps $$\mu\mapsto\sqrt{\KL(\mu\p\nu)}$$ and $$\nu\mapsto\sqrt{\KL(\mu\p\nu)}$$ are not convex in general. Indeed, ...
Iosif Pinelis's user avatar
9 votes

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

In fact one can say quite a lot of regularity of the boundary of a convex set. Assume that $X\subset\mathbb{R}^n$ is a bounded convex set with non-empty interior. Convex functions are locally ...
Piotr Hajlasz's user avatar
9 votes
Accepted

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

$\newcommand{\de}{\delta} \newcommand{\De}{\Delta} \newcommand{\ep}{\varepsilon}$ Take any probability measures $P_0,P_1$ absolutely continuous with respect (w.r.) to $Q$. We shall prove the ...
Iosif Pinelis's user avatar
9 votes

Elementary inequality generalizing convexity of a function on a segment

Here's another argument using Sturm-Liouville theory. Given $b - \pi < a \le x \le b$, let $$ f(x) = \sin (x - b + \pi). $$ Observe that $f > 0$ and $f'' + f = 0$ on $[a,b)$, $f(a) > 0$, $f(...
Deane Yang's user avatar
9 votes

Convexity and Lipschitz continuity

Yes Consider first the case where $f\in{\cal C}^2$. Then $$\nabla f(y)-\nabla f(x)=\int_0^1{\rm D}^2f(x+t(y-x))\cdot(y-x)\,dt.$$ There follows $$\|\nabla f(y)-\nabla f(x)\|\le\|y-x\|\int_0^1\|{\rm D}^...
Denis Serre's user avatar
  • 51.7k
9 votes
Accepted

Convergence of convex functions

It follows from Theorem 10.8 in R. Tyrrell Rockafellar. Convex analysis. Princeton Mathematical Series, No. 28. Princeton University Press, Princeton, N.J., 1970. This theorem essentially says the ...
Del's user avatar
  • 380
8 votes
Accepted

Reference for Minkowski functional when 0 is not in the interior

I will try to give a concise answer here (omitting the proofs) and leave some references in the end. There will be three parts in this answer, the first two rather introductory. Throughout this answer,...
Ivan Solonenko's user avatar
8 votes

Convergence of convex functions

I hope it's OK to post with a proof rather than a reference; when I started writing I thought this would be shorter... Lemma. Let $(f_n \mid n \geq 1)$ be a sequence of convex functions $f_n: \mathbf{...
Leo Moos's user avatar
  • 4,968
8 votes

Conditions for including cones

$\newcommand\R{\mathbb R}$Update: Within the previously established framework, we now show that it is enough to have just $N=4$ points. This improves what seems to be the previous record, of $N=5$. It ...
Iosif Pinelis's user avatar
8 votes
Accepted

A potential new norm for matrices and Horn's inequalities

By the standard classification of unitarily invariant norms (see e.g., this blog post), the expression $N_{(p,q)}$ is a norm if and only if the function $$ \| (x,y) \| := \inf \left\{ \varepsilon > ...
Terry Tao's user avatar
  • 110k
7 votes

Lipschitz function admits Whitney stratification

In general no such stratification exists. Lipschitz functions coincide with $C^1$ functions on sets of positive measure: Theorem. If $f$ is Lipschitz in $\mathbb{R}^n$, then for any $\epsilon>0$...
Piotr Hajlasz's user avatar
7 votes
Accepted

Convex function and weak convergence of measures

Let $\mathscr{X}$ be a locally convex space and $X \subset \mathscr{X}$ compact. Let $E = \{f \in C(X) : \langle f, \mu_n \rangle \to \langle f, \mu \rangle\}$. It's easy to see that $E$ is a closed ...
Nate Eldredge's user avatar

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