44
votes

### Strange result about convexity

The inequality you wrote down is a special case of a general principle about higher-order convexity: any "simple enough" linear inequality of the type you wrote down will be true as long as ...

38
votes

Accepted

### Why the sequence of Bernstein polynomials of $\sqrt x$ is increasing?

As noted by Paata Ivanishvili, if $f$ is concave on $[0,1]$, then the Bernstein polynomials $B_n(f,p)$ are increasing in $n$.
Here is a probabilistic proof:
Let $I_j$ for $j \ge 1$ be independent ...

30
votes

### Nonconvexity and discretization

I was asked to answer this question. The people one really needs to ask here are the people in high-dimensional convex geometry/probability/statistics/computer science and metric geometry, but in any ...

29
votes

Accepted

### Reference to a conjecture on unit vectors in Euclidean space

That isn't a conjecture but a routine exercise assigned after the students learn about Bang's solution of the Tarski plank problem. The proof goes in 2 steps:
1) Consider all sums $\sum_j \...

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 ...

23
votes

### Nonconvexity and discretization

I will start with a reformulation of the Proposition in the question in terms that are easier to digest for analysts.
Proposition: let $0<\tilde{r}<r<1$ and $0<p<1$ be such that $\tilde{...

23
votes

### Functions of $\mathbb{R}^d$ preserving convexity of sets

See Meyer, Walter; Kay, David C., A convexity structure admits but one real linearization of dimension greater than one, J. Lond. Math. Soc., II. Ser. 7, 124-130 (1973). ZBL0271.52002.
Theorem 4. If $...

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 ...

21
votes

### The missing link: an inequality

Fix $n$, and let $V(x)$ be the polynomial from the question, for which we want to show that it is
positive on the open interval $(0,1)$. Set
\begin{align*}
a(n) &= 1024n^2 - 4096/3n + 320\\
b(n) &...

20
votes

### Why the sequence of Bernstein polynomials of $\sqrt x$ is increasing?

While nothing will beat the brilliant probabilistic proof given in Yuval Peres's answer, a more conventional argument goes as follows. Write $$a_{n,k} = \tbinom nk x^k (1-x)^{n-k} $$ and $$ p_{n,k} = \...

Community wiki

17
votes

### Are small $\varepsilon$-balls convex in geodesic metric spaces?

The standard sub-Riemannian metric on the 3-dimensional real Heisenberg group $\mathbb{H}^3$ is a counterexample. Topologically, it is homemorphic to $\mathbb{R}^3$, so in particular it is locally ...

17
votes

### How to make a sandwich from just one piece of bread?

This won't be a complete answer to Q2, but something of a starting point, at least for all two-dimensional convex $P$. Assuming for simplicity that the area of $P$ is 1, $P$ contains a parallelogram ...

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)...

16
votes

### Why the sequence of Bernstein polynomials of $\sqrt x$ is increasing?

I would also be happy to know the converse implication, that you quoted in the last remark of your notes
Let $f \in C([0,1])$. Then $p_{n+1}(f,x) \geq p_{n}(f,x)$ for all $n \in \mathbb{N}$ and all $x ...

15
votes

Accepted

### Strange result about convexity

The answer is yes, and it follows immediately from a simple description of the extreme rays of the convex cone of the functions with a convex second derivative.
Indeed, for $x\in(0,1]$, let $f'''(x)$ ...

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 ...

12
votes

Accepted

### weak*-closed convex = closed convex?

No it is false in general.
Yes, there is a class of spaces where it is true: these are exactly the reflexive spaces.
Suppose $X$ is not reflexive. Then considering $X$ embedded into $X''$, we have $...

12
votes

Accepted

### Are small $\varepsilon$-balls convex in geodesic metric spaces?

Negative part
The answer is no without further assumptions, here is a counterexample (a bit nasty, it is not locally simply connected) :
In the plane consider first the two positive semi axis. ($\...

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

### A square root inequality for symmetric matrices?

The classical operator generalization of the scalar inequality $|\sqrt{a}-\sqrt{b}|^2 \leq |a-b|$ is the Powers-Størmer inequality, which involves two different norms : the trace norm $\|X\|_1 = \...

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

### Simple-looking problem with integrals

As suggested by Fedor Petrov, we write
$$ g(x) = f(\sqrt x) , $$
and we substitute $\lambda \sin\theta = \sqrt{x}$ and $t = \lambda^2$. This leads to
$$ \begin{aligned} 0 & = 2 \lambda \int_0^{\pi/...

11
votes

### The missing link: an inequality

This proof is mainly based on some of the suggestions in the early version of this answer. Also, we are building here on the comment by Peter Mueller concerning the polynomial $a$ defined below. It ...

11
votes

### How to make a sandwich from just one piece of bread?

Restricting to the case where $P$ is triangular, if the angles of the triangle are $A$, $B$, $C$ and you fold the triangle along the bisector to the angle $C$, then assuming wlog that $A<B$, you ...

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

### mixing convex and concave for convexity

Time to start typing. As I said, this is going to be long and boring and I have only limited amount of free time nowadays so I'll type in chunks. If you see a flaw somewhere, comment immediately, but ...

10
votes

### Volumes of sets of constant width in high dimensions

The question has now been solved by Andrii Arman, Andriy Bondarenko, Fedor Nazarov, Andriy Prymak, and Danylo Radchenko. The answer is yes! there are sets of constant width
with volume less than $0.9^...

10
votes

### The missing link: an inequality

I sketch a method that should show $F_n$ is convex for $n$ sufficiently large.
A Taylor expansion gives that $F_n(x) = x^{2n} (1-x)^2 + O(x^{4n-1})$, and so $$F_n''(x) = 2x^{2n-2}[2n^2(1-x)^2 + x^2 - ...

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

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