A note on the extreme points of the cone of quasiconvex quadratic forms with orthotropic symmetry

TL;DR

This paper investigates the extreme points of the cone of quasiconvex quadratic forms with orthotropic symmetry, linking extremality of associated polynomials to the extremality of quadratic forms within the same class.

Contribution

It establishes a novel connection between extremal polynomials and the extremality of quadratic forms in the context of orthotropic symmetry.

Findings

Extremal polynomials imply the quadratic form is an extreme point.

The study characterizes extremality within the cone of quasiconvex quadratic forms.

Results are grounded in classical convex analysis.

Abstract

We study the extreme points of the cone of quasiconvex quadratic forms with linear elastic orthotropic symmetry. We prove that if the determinant of the acoustic matrix of the associated forth order tensor of the quadratic form is an extremal polynomial, then the quadratic form is an extreme point of the cone in the same symmetry class. The extremality of polynomials and quadratic forms here is understood in classical convex analysis sense.