The Algebraic Boundary of SO(2)-Orbitopes

TL;DR

This paper characterizes the algebraic boundary of SO(2)-orbitopes, identifying when secant varieties form boundary components, and applies this to determine the basic closedness of specific orbitopes.

Contribution

It provides a characterization of the algebraic boundary of SO(2)-orbitopes and identifies all boundary components for 4-dimensional cases, advancing understanding of their semi-algebraic properties.

Findings

The r-th secant variety is an irreducible component of the algebraic boundary under certain conditions.

All irreducible components of the algebraic boundary for 4-dimensional SO(2)-orbitopes are identified.

Conditions for these orbitopes to be basic closed semi-algebraic sets are established.

Abstract

Let $X\subset\A^{2r}$ be a real curve embedded into an even-dimensional affine space. In the main result of this paper, we characterise when the r-th secant variety to $X$ is an irreducible component of the algebraic boundary of the convex hull of the real points $X(\R)$ of $X$. This fact is then applied to 4-dimensional SO(2)-orbitopes and to the so called Barvinok-Novik orbitopes to study when they are basic closed as semi-algebraic sets. In the case of 4-dimensional SO(2)-orbitopes, we find all irreducible components of their algebraic boundary.