The Geometry of Discotopes

TL;DR

This paper explores the geometric and algebraic properties of discotopes, a class of convex bodies related to zonoids, focusing on their face structure, boundary hypersurfaces, and birational properties using algebraic geometry tools.

Contribution

It introduces the study of discotopes' face structure and boundary hypersurfaces, revealing birational properties and providing degree bounds for hypersurfaces in special cases.

Findings

Zariski closure of extreme points forms an irreducible hypersurface for certain discotopes.

An upper bound for the degree of the boundary hypersurface is established.

Connections to classical determinantal varieties are identified.

Abstract

We study a class of semialgebraic convex bodies called discotopes. These are instances of zonoids, objects of interest in real algebraic geometry and random geometry. We focus on the face structure and on the boundary hypersurface of discotopes, highlighting interesting birational properties which may be investigated using tools from algebraic geometry. When a discotope is the Minkowski sum of two-dimensional discs, the Zariski closure of its set of extreme points is an irreducible hypersurface. In this case, we provide an upper bound for the degree of the hypersurface, drawing connections to the theory of classical determinantal varieties.