The Logarithmic Minkowski Problem

Combinatorics and more 2021-11-23

NYC-origin

The logarithmic origin of Manhattan

We are spending the fall semester in NYC at NYU and yesterday* I went to lunch with two old friends Deane Yang and Gaoyong Zhang. They told me about the logarithmic Minkowski problem, presented in the paper The logarithmic Minkowski problem by Károly Böröczky, Erwin Lutwak, Deane Yang and Gaoyong Zhang (BLYZ). (See also this paper by BLYZ.) We will get to the problem after a short reminder of Minkowski’s theorem.

The Discrete Minkowski problem: Find necessary and sufficient conditions on a set of unit vectors $u_1,\dots, u_m$ in $\mathbb R^n$ and a set of real numbers $a_1,\dots,a_m > 0$ that will guarantee the existence of an $m$ -faced polytope in $\mathbb R^n$ whose faces have outer unit normals $u_1,\dots, u_m$ and corresponding face-areas $a_1,\dots,a_m$ .

Minkowski himself gave a complete solution in 1911, a necessary and sufficient condition is that the following relation holds:

$a_1u_1 + a_2u_2 + \cdots a_mu_m=0$ .

If a polytope contains the origin in its interior, then the cone-volume associated with a face of the polytope is the volume of the convex hull of the face and the origin.

The Discrete logarithmic Minkowski problem: Find necessary and sufficient conditions on a set of unit vectors $u_1,\dots, u_m$ in $\mathbb R^n$ and a set of real numbers $a_1,\dots,a_m > 0$ that will guarantee the existence of an $m$ -faced polytope in $\mathbb R^n$ whose faces have outer unit normals $u_1,\dots, u_m$ and corresponding cone-volumes $a_1,\dots,a_m$ .

The problem was solved by BLYZ for the centrally symmetric case. It is related to a lot of deep mathematics (convexity, valuations, Brunn-Minkowski theory, analysis, PDE… perhaps combinatorics). In both cases there are continuous versions that are a little harder to formulate.

20211102_134826

With Deane Yang and Gaoyong Zhang

* Actually it was four weeks ago and since then we had another lunch and Deane explained some recent issues that are discussed/debated in the American mathematical community with a lot of excitement. (I did not really understand.) (Actually there was a thorough and nice discussion on Facebook a year ago about double-blind refereeing which seems a small corner of the large excitement.)