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# Tag Info

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 ...
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 ...
Accepted

### 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) &...

### 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 ...
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 ...
Accepted

### 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 - ...

### 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 ...
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 $C:=\{x:\forall\xi\in \mathbb{R}^n,\langle \xi,x\rangle\leqslant h(\xi)\}$. ...
The following is the completely monotonic claim that actually holds (also hinted by Iosif Pinelis). Claim. Let $f(x)=\log\binom{x}{px}$; then, $f''$ is CM. We prove this claim as a corollary of ...