Algebraic Boundaries of Convex Semi-algebraic Sets

TL;DR

This paper explores the algebraic boundaries of convex semi-algebraic sets using duality, generalizing classical polytope duality, characterizing exceptional points, and providing an algorithm for their computation.

Contribution

It extends duality concepts to general semi-algebraic convex bodies, characterizes exceptional extreme points, and introduces an algorithm to compute these points from the algebraic boundary.

Findings

Generalized polytope duality to semi-algebraic convex sets

Characterized semi-algebraic exceptional extreme points

Provided an algorithm to compute exceptional families

Abstract

We study the algebraic boundary of a convex semi-algebraic set via duality in convex and algebraic geometry. We generalize the correspondence of facets of a polytope to the vertices of the dual polytope to general semi-algebraic convex bodies. In the general setup, exceptional families of extreme points might exist and we characterize them semi-algebraically. We also give an algorithm to compute a complete list of exceptional families, given the algebraic boundary of the dual convex set.