Abstract

Let distK\dist_K be the distance from a compact set K⊂Mm×nK\subset\mathbb{M}^{m\times n} in the space of m×nm\times n matrices. This note determines the set Mp⊂Mm×nM_p\subset \mathbb{M}^{m\times n} of zeros of the polyconvex hull of distKp\dist_K^p where 1≤p<∞1\leq p<\infty. It is shown that the set-valued map p↦Mpp\mapsto M_p is constant on the intervals [1,2),…,[q−1,q),[q,∞)[1,2),\dots,[q-1,q),[q,\infty) where q:=min⁡{m,n}q:=\min\{ m, n\}, while at p=1,…,qp=1,\dots,q the set MpM_p generally jumps down discontinuously. The values MsM_s, s=1,…,qs= 1,\dots,q, at the beginnings of intervals of constancy are characterized as ss-polyconvex hulls PsK\P^sK of KK to be defined below, where P1K\P^1K is the convex hull and PqK\P^qK the standard polyconvex hull. As an illustration, PsSO(n)\P^sSO(n) are evaluated for all ss if 1≤n≤41\leq n\leq 4, and for nn arbitrary if n≥s>n/2n\geq s>n/2 and/or s=1s=1. In the remaining cases only bounds are obtained.

Author information

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Miroslav Šilhavý

Dip. di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy
Permanent Address: Mathematical Institute, Academy of Sciences, Žitná 25, 115 67 Prague 1, Czech Republic

silhavy@math.cas.cz

Keywords

  • Semiconvexity
  • polyconvexity
  • polyconvex hulls
  • rotational invariance

Mathematics Subject Classification

49J45; 74B20

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Suggested citation

M. Šilhavý. “Zeros of the Polyconvex Hull of Powers of the Distance and s-Polyconvexity.” Journal of Convex Analysis 14 (2007), No. 2, 319–344. https://doi.org/10.68381/jca14022

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Published by Heldermann Verlag, 2007. Rights now held by Banach Press LLC.