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

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].