Journal of Convex Analysis, Vol. 6, No. 1, pp. 207-213 (1999)

On the composition of quasiconvex functions and the transposition

Martin Kruzik

Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, Pod vodarenskou vezi 4, 182 08 Praha 8, Czech Republic,

Abstract: If $G:\mathbb{R}^{n\times m}\to\bar\mathbb{R}:=\mathbb{R}\cup\{+\infty\}$ is a convex, polyconvex or rank-one convex function, then the function $g:\mathbb{R}^{m\times n}\to\bar\mathbb{R}$ defined as $g(A)=G(A^t)$ preserves convexity, polyconvexity, or rank-one convexity, respectively. The paper shows that this does not hold in general for quasiconvexity provided $n\ge 2$ and $m\ge 3$.

Keywords: Polyconvexity, quasiconvexity, rank-one convexity

Classification (MSC2000): 26B25, 49R99

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