This article is cited in 1 paper

The convex hull and the Carathéodory number of a set in terms of the metric projection operator

K. S. Shklyaev<sup>ab

a</sup> Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia
^b Moscow Center of Fundamental and Applied Mathematics, Moscow, Russia

Abstract: We prove that each point of the convex hull of a compact set $M$ in a smooth Banach space $X$ can be approximated arbitrarily well by convex combinations of best approximants from $M$ to $x$ (values of the metric projection operator $P_M(x)$), where $x \in X$. As a corollary, we show that the Carathéodory number of a compact set $M \subset X$ with at most $k$-valued metric projection $P_M$ is majorized by $k$, that is, each point in the convex hull of $M$ lies in the convex hull of at most $k$ points of $M$.
Bibliography: 26 titles.

Keywords: metric projection, convex hull, Banach space, smoothness, Minkowski functional, Carathéodory number.

MSC: Primary 41A65; Secondary 52A35, 52A20

Received: 23.02.2022 and 11.05.2022

DOI: 10.4213/sm9742

 English version: Sbornik: Mathematics, 2022, 213:10, 1470–1486

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