Two results on the size of spectrahedral descriptions

TL;DR

This paper investigates the minimal matrix size needed to represent convex sets as spectrahedra, providing lower bounds for various geometric cases including the unit ball and certain algebraic boundaries.

Contribution

It establishes new lower bounds on the size of matrices in spectrahedral descriptions for specific convex sets and algebraic boundaries.

Findings

For the n-dimensional unit ball, the minimal size r is at least n/2.

When n=2^k+1, the minimal size r equals n.

For certain cubic algebraic boundaries in R^3, r is at least five.

Abstract

A spectrahedron is a set defined by a linear matrix inequality. Given a spectrahedron we are interested in the question of the smallest possible size $r$ of the matrices in the description by linear matrix inequalities. We show that for the $n$-dimensional unit ball $r$ is at least $\frac{n}{2}$. If $n=2^k+1$, then we actually have $r=n$. The same holds true for any compact convex set in $\mathbb{R}^n$ defined by a quadratic polynomial. Furthermore, we show that for a convex region in $\mathbb{R}^3$ whose algebraic boundary is smooth and defined by a cubic polynomial we have that $r$ is at least five. More precisely, we show that if $A,B,C$ are real symmetric matrices such that $f(x,y,z)=\det(I+A x+B y+C z)$ is a cubic polynomial, the surface in complex projective three-space with affine equation $f(x,y,z)=0$ is singular.