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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

$f(A)=\{\begin{array}{ll}\frac{1}{\det (A)},& \det (A)>0;\\ +\mathrm{\infty },& \det (A)\le 0;\end{array}$

is polyconvex but not convex.

References

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