Jesús A. De Loera, Ethan X. Fang, Shengtao Guo, Junwei Lu, and Hailun Zheng Proved the Simplex–Cube Conjecture for Simple Polytopes.

Shana Tova Shana Tova (happy new Jewish year) to all our readers! We have just returned to Tel Aviv from the beautiful city of Tiberias, on the Sea of Galilee. The Simplex cube conjecture The simplex–cube conjecture was posed in my 1990 paper and was among the five problems on convex polytopes discussed in this 2008 post. Conjecture A. For every there exists an integer such that if is a -polytope with , then has a -face which is either a simplex or (combinatorially) a cube. We denote by the smallest such integer, if it exists, and otherwise set . A weaker conjecture, which remains open in general, is the following. Conjecture B. For every positive integer , there exist an integer and a finite collection of -dimensional polytopes such that every -polytope with has a -face combinatorially equivalent to a member of . As with , we let denote the smallest possible threshold, and set if no such threshold exists. Euler’s theorem implies that $latex d′(2)=3$: every 3-polytope has a 2-face…