Polyconvex function

In mathematics, the notion of polyconvexity is a generalization of the notion of convexity for functions defined on spaces of matrices. Let "M"_{"m"×"n"}("K") denote the space of all "m" × "n" matrices over the field "K", which may be either the real numbers R or the complex numbers C. A function "f" : "M"_{"m"×"n"}("K") → R ∪ {±∞} is said to be polyconvex if

:$A\; mapsto\; f(A)$

can be written as a convex function of the "p" × "p" subdeterminants of "A", for 1 ≤ "p" ≤ min{"m", "n"}.

Polyconvexity is a weaker property than convexity. For example, the function "f" given by

:

is polyconvex but not convex.

References

* cite book
author = Renardy, Michael and Rogers, Robert C.
title = An introduction to partial differential equations
series = Texts in Applied Mathematics 13
edition = Second edition
publisher = Springer-Verlag
location = New York
year = 2004
pages = 353
id = ISBN 0-387-00444-0 (Definition 9.25)