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 \\...

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 ...

22
votes

Accepted

### 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 ...

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 ...

16
votes

Accepted

### 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)...

15
votes

Accepted

### 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 ...

13
votes

Accepted

### 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 ...

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. ...

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 $...

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 ...

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 $C^1$...

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" ...

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 ...

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 ...

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} ...

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 $C:=\{x:\forall\xi\in \mathbb{R}^n,\langle \xi,x\rangle\leqslant h(\xi)\}$. ...

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 ...

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, ...

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 ...

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 ...

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(...

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}^...

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 ...

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,...

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{...

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 ...

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 > ...

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 $\epsilon>0$...

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 ...

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