Polyconvex function
From HandWiki
In mathematics, the notion of polyconvexity is a generalization of the notion of convexity for functions defined on spaces of matrices. Let Mm×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 : Mm×n(K) → R ∪ {±∞} is said to be polyconvex if
- [math]\displaystyle{ A \mapsto f(A) }[/math]
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
- [math]\displaystyle{ f(A) = \begin{cases} \frac1{\det (A)}, & \det (A) \gt 0; \\ + \infty, & \det (A) \leq 0; \end{cases} }[/math]
is polyconvex but not convex.
References
- Renardy, Michael; Rogers, Robert C. (2004). An introduction to partial differential equations. Texts in Applied Mathematics 13 (Second ed.). New York: Springer-Verlag. pp. 353. ISBN 0-387-00444-0. (Definition 10.25)
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