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

TL;DR

This paper characterizes the extreme rays of the cone of 3x3 quasiconvex quadratic forms by analyzing the extremality of the determinant of their acoustic tensor, revealing new classifications and conjectures for these forms.

Contribution

It provides a complete characterization of extreme rays based on the extremality of the acoustic tensor's determinant, including new results and conjectures for 3x3 quasiconvex quadratic forms.

Findings

Forms with extremal non-square determinant are extreme rays.

Forms with perfect square determinant are either extreme rays or polyconvex.

Zero determinant forms are polyconvex.

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 ${\cal C}_3$). We characterize quadratic forms $f\in {\cal C}_3,$ 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 $f\in {\cal C}_3$ is an extremal polynomial other than a perfect square, then the form must itself be an extreme ray of ${\cal C}_3;$ when the determinant is a perfect square, then the form is either an extreme ray of ${\cal C}_3$ 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 ${\cal C}_3,$ and discuss about weak and strong etremals of ${\cal C}_d$ for $d\geq 3.$ where it turns out that several properties of ${\cal C}_3$ do not hold for ${\cal C}_d$ 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 ${\cal C}_3$ that were proved to be such. Our results also improve the ones proven by the first author and Milton [20].