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 \\...
- 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 ...
- 7,463
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 ...
- 12.6k
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 ...
- 66.8k
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)...
- 51.6k
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 ...
- 59.8k
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 ...
- 88.9k
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. ...
- 4,950
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 $...
- 4,792
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 ...
- 16.3k
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 $C^1$...
- 12.3k
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" ...
- 5,755
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 ...
- 12.3k
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 ...
- 37.6k
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} ...
- 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 $C:=\{x:\forall\xi\in \mathbb{R}^n,\langle \xi,x\rangle\leqslant h(\xi)\}$. ...
- 103k
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 ...
- 59.8k
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, ...
- 118k
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 ...
- 27.3k
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 ...
- 118k
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(...
- 27k
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
$$\|\nabla f(y)-\nabla f(x)\|\le\|y-x\|\int_0^1\|{\rm D}^...
- 51.6k
9
votes
Accepted
Convergence of convex functions
It follows from Theorem 10.8 in
R. Tyrrell Rockafellar. Convex analysis. Princeton Mathematical Series, No.
28. Princeton University Press, Princeton, N.J., 1970.
This theorem essentially says the ...
- 380
8
votes
Accepted
Reference for Minkowski functional when 0 is not in the interior
I will try to give a concise answer here (omitting the proofs) and leave some references in the end. There will be three parts in this answer, the first two rather introductory. Throughout this answer,...
- 538
8
votes
Convergence of convex functions
I hope it's OK to post with a proof rather than a reference; when I started writing I thought this would be shorter...
Lemma. Let $(f_n \mid n \geq 1)$ be a sequence of convex functions $f_n: \mathbf{...
- 4,968
8
votes
Conditions for including cones
$\newcommand\R{\mathbb R}$Update: Within the previously established framework, we now show that it is enough to have just $N=4$ points. This improves what seems to be the previous record, of $N=5$. It ...
- 118k
8
votes
Accepted
A potential new norm for matrices and Horn's inequalities
By the standard classification of unitarily invariant norms (see e.g., this blog post), the expression $N_{(p,q)}$ is a norm if and only if the function
$$ \| (x,y) \| := \inf \left\{ \varepsilon > ...
- 109k
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 ...
- 29.2k
7
votes
An intuition for three different types of subgradients (proximal, regular, limiting)
This might not be a satisfying answer, but this is how I personally deal with this issue (at least in the topic of optimization).
I think these concepts are not made for actually computing them but ...
- 181
Only top scored, non community-wiki answers of a minimum length are eligible