On the extreme rays of the cone of $3\times 3$ quasiconvex quadratic forms: Extremal determinats vs extremal and polyconvex forms

Abstract

This work is concerned with the study of the extreme rays of the convex cone of $3\times 3$ quasiconvex quadratic forms (denoted by ${\mathcal{C}}_3$). We characterize quadratic forms $f\in {\mathcal{C}}_3,$ the determinant of the acoustic tensor of which is an extremal polynomial, and conjecture/discuss about other cases. We prove that in the case when the determinant of the acoustic tensor of a form $f\in {\mathcal{C}}_3$ is an extremal polynomial other than a perfect square, then the form must itself be an extreme ray of ${\mathcal{C}}_3;$ when the determinant is a perfect square, then the form is either an extreme ray of ${\mathcal{C}}_3$ or polyconvex; and finally, when the determinant is identically zero, then the form $f$ must be polyconvex. The zero determinant case plays an important role in the proofs of the other two cases. We also make a conjecture on the extreme rays of ${\mathcal{C}}_3,$ and discuss about weak and strong etremals of ${\mathcal{C}}_d$ for $d\ge 3.$ where it turns out that several properties of ${\mathcal{C}}_3$ do not hold for ${\mathcal{C}}_d$ for $d>3,$ and thus case $d=3$ is special. These results recover all previously known results (to our best knowledge) on examples of extreme points of ${\mathcal{C}}_3$ that were proved to be such. Our results also improve the ones proven by the first author and Milton [20].

25 pages. Mistake on page 20 fixed

References in corpus (2)

• On Hilbert's construction of positive polynomials
• Sums of Squares: A Real Projective Story

Cited by in corpus (1)

• (Positive) Quadratic Determinantal Representations of Quartic Curves and the Robinson Polynomial