A Topological Approach to Simple Descriptions of Convex Hulls of Sets Defined by Three Quadrics

TL;DR

This paper introduces a topological method to analyze the convex hulls of sets defined by three quadratic inequalities, revealing new aggregation families and characterizing the emptiness of related projective varieties.

Contribution

It develops a novel spectral curve approach linking aggregations to convex hull descriptions and generalizes existing results on the emptiness of quadratic varieties.

Findings

Identifies new families of convex hulls defined by aggregations.

Provides a spectral curve-based characterization of the emptiness of quadratic varieties.

Extends previous results on the spectral properties of quadratic inequalities.

Abstract

We study the convex hull of a set $S\subset \mathbb{R}^n$ defined by three quadratic inequalities. A simple way of generating inequalities valid on $S$ is to take nonnegative linear combinations of the defining inequalities of $S$. We call such inequalities aggregations. We introduce a new technique relating aggregations to properties of the spectral curve, i.e. the curve defined by the vanishing of the determinant polynomial, and utilizing known spectral sequences (Agrachev and Lerario, 2012). We find new families beyond those identified in (Dey, Mu\~noz, and Serrano, 2022; Blekherman, Dey, and Sun, 2024), where the convex hull is defined by aggregations. We also prove a characterization of the emptiness of the projective variety defined by $3$ homogeneous quadratics in terms of the spectral curve generalizing results of (Agrachev, 1988).