# Tag Info

### 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
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 ...
• 7,578
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 ...
• 12.7k
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### 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 ...
• 67.4k
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• 4,802
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### 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
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### 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 $C^1$...
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### 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" ...
• 5,765
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 ...
• 12.4k
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### 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 ...
• 38k
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### 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
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 $C:=\{x:\forall\xi\in \mathbb{R}^n,\langle \xi,x\rangle\leqslant h(\xi)\}$. ...
• 104k
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### 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 $\epsilon>0$...
• 27.4k
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 ...