Abstract

We consider a certain metric on the space of all convex compacta in Rn\mathbb{R}^n, introduced by A. Plis ["Uniqueness of optimal trajectories for non-linear control problems, Ann. Polon. Math. 29 (1975), 397-401]. The set of strictly convex compacta is a complete metric subspace of the metric space of convex compacta with respect to this metric. We present some applications of this metric to the problems of set-valued analysis, in particular we estimate the distance between two compact sets with respect to this metric and to the Hausdorff metric.

Author information

Contact details are reproduced from the original publication and may be historical.

Maxim Viktorovich Balashov

Dept. of Higher Mathematics, Moscow Institute of Physics and Technology, Institutski str. 9, Dolgoprudny, Moscow Region, Russia 141700

balashov@mail.mipt.ru

Dušan Repovš

Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia

dusan.repovs@guest.arnes.si

Keywords

  • Metric space
  • strictly convex compactum
  • modulus of convexity
  • set-valued mapping
  • strict convexity
  • uniform convexity
  • supporting function
  • Demyanov distance
  • Hausdorff distance

Mathematics Subject Classification

54A20, 52A41; 52A20, 52A99, 46N10

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Suggested citation

M. V. Balashov, D. Repovš. “On Pliś Metric on the Space of Strictly Convex Compacta.” Journal of Convex Analysis 19 (2012), No. 1, 171–183. https://doi.org/10.68381/jca19010

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Published by Heldermann Verlag, 2012. Rights now held by Banach Press.