The convex hull and the Carath\'eodory number of a~set in terms of the metric projection operator
Sbornik. Mathematics, Tome 213 (2022) no. 10, pp. 1470-1486
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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.
@article{SM_2022_213_10_a6,
author = {K. S. Shklyaev},
title = {The convex hull and the {Carath\'eodory} number of a~set in terms of the metric projection operator},
journal = {Sbornik. Mathematics},
pages = {1470--1486},
publisher = {mathdoc},
volume = {213},
number = {10},
year = {2022},
language = {en},
url = {http://geodesic.mathdoc.fr/item/SM_2022_213_10_a6/}
}
TY - JOUR AU - K. S. Shklyaev TI - The convex hull and the Carath\'eodory number of a~set in terms of the metric projection operator JO - Sbornik. Mathematics PY - 2022 SP - 1470 EP - 1486 VL - 213 IS - 10 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/SM_2022_213_10_a6/ LA - en ID - SM_2022_213_10_a6 ER -
K. S. Shklyaev. The convex hull and the Carath\'eodory number of a~set in terms of the metric projection operator. Sbornik. Mathematics, Tome 213 (2022) no. 10, pp. 1470-1486. http://geodesic.mathdoc.fr/item/SM_2022_213_10_a6/