Displaying similar documents to “A complete characterization of invariant jointly rank-r convex quadratic forms and applications to composite materials”

A note on equality of functional envelopes

Martin Kružík (2003)

Mathematica Bohemica

Similarity:

We characterize generalized extreme points of compact convex sets. In particular, we show that if the polyconvex hull is convex in ℝ m × n , min ( m , n ) ≤ 2 , then it is constructed from polyconvex extreme points via sequential lamination. Further, we give theorems ensuring equality of the quasiconvex (polyconvex) and the rank-1 convex envelopes of a lower semicontinuous function without explicit convexity assumptions on the quasiconvex (polyconvex) envelope.

On semiconvexity properties of rotationally invariant functions in two dimensions

Miroslav Šilhavý (2004)

Czechoslovak Mathematical Journal

Similarity:

Let f be a function defined on the set 𝐌 2 × 2 of all 2 by 2 matrices that is invariant with respect to left and right multiplications of its argument by proper orthogonal matrices. The function f can be represented as a function f ˜ of the signed singular values of its matrix argument. The paper expresses the ordinary convexity, polyconvexity, and rank 1 convexity of f in terms of its representation f ˜ .

Rank 1 convex hulls of isotropic functions in dimension 2 by 2

Miroslav Šilhavý (2001)

Mathematica Bohemica

Similarity:

Let f be a rotationally invariant (with respect to the proper orthogonal group) function defined on the set M 2 × 2 of all 2 by 2 matrices. Based on conditions for the rank 1 convexity of f in terms of signed invariants of 𝔸 (to be defined below), an iterative procedure is given for calculating the rank 1 convex hull of a rotationally invariant function. A special case in which the procedure terminates after the second step is determined and examples of the actual calculations are given. ...