Approximation of a symmetric convex body by polytopes

Let be an origin-symmetric symmetric convex body in , What is the optimal upper bound for the number of vertices of the convex polytope satisfying ? Here we may consider both large and small . As far as I understand, in full generality the 2-parametric asymptotic is still open, see some recent results and a …

Euler’s proof of pentagonal theorem

tetrahedron

Hi there! Today we prove that in an arbitrary tetrahedron . Draw three planes: through orthogonal to the external bisector of , through orthogonal to the external bisector of , through orthogonal to the external bisector of . Let them meet at point , and let be projections of onto lines respectively (see the picture). …

e⩽π

Mathematicians do like giving proofs of known results from some specific point of view. Here goes the combinatorial proof of the inequality To be precise, I define our heroes. is a length of a half-circle of the unit radius, and is the common point of the segments The idea is that for a positive integer …

Low-level variant of Huang’s argument for sensitivity conjecture

Consider the set of all words , over the alphabet (the vertices of Boolean cube). We naturally multiply the words by concatenation: if , then . For and denote by the word which differs from only in -th position. Clearly the operators are involutive and mutually commute. We further use the same notations for different …

Welcome to My New Blog

Hi all! I finally decided to have some blog with mathematical notes (I do not want to publish anything not concerning mathematics here, in particular this is why everything is assumed to be in “English”). I would continue on rus4.livejournal.com, but unfortunately livejournal does not support formulae and the easiest way to fix it seems …