Two Results on the Size of Spectrahedral Descriptions

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 $A,B,C \in {Sym}_r(\mathbb{R})$ are real symmetric matrices such that $f(x,y,z)=\det(I_r+A x+B y+C z)$ is a cubic polynomial, then the surface in complex projective three-space with affine equation $f(x,y,z)=0$ is singular.