Families of faces and the normal cycle of a convex semi-algebraic set

TL;DR

This paper explores the geometric structure of convex semi-algebraic sets through the normal cycle, introducing a new algebraic concept called a 'patch' for approximating convex hulls and analyzing face families.

Contribution

It introduces the notion of a 'patch' within convex algebraic geometry and studies the normal cycle for convex semi-algebraic sets, linking face families to projective duality.

Findings

Normal cycle is a semi-algebraic set similar to the conormal variety.

The 'patch' concept aids in approximating convex hulls of semi-algebraic sets.

Geometric implications for semi-algebraic and convex geometry are discussed.

Abstract

We study families of faces for convex semi-algebraic sets via the normal cycle which is a semi-algebraic set similar to the conormal variety in projective duality theory. We propose a convex algebraic notion of a "patch" -- a term recently coined by Ciripoi, Kaihnsa, L\"ohne, and Sturmfels as a tool for approximating the convex hull of a semi-algebraic set. We discuss geometric consequences, both for the semi-algebraic and convex geometry of the families of faces, as well as variations of our definition and their consequences.