Each simple convex polytope from $\mathbb{R}^n$ has a point with $2n+4$ normals to the boundary

I. Nasonov^a, G. Panina<sup>b

a</sup> Saint Petersburg University, 7/9 Universitetskaya nab., St. Petersburg, 199034 Russia
^b St. Petersburg department of Steklov institute of mathematics RAS, Fontanka 27, St. Petersburg, 191023 Russia

Abstract: We prove that for $n>3$ each generic simple polytope in $\mathbb{R}^n$ contains a point with at least $2n+4$ emanating normals to the boundary. This result is a piecewise-linear counterpart of a long-standing problem about normals to smooth convex bodies.

Key words and phrases: Morse theory, bifurcation, critical point.

UDC: 514.172.45

Received: 01.12.2025

Language: English