# MR2580162 (2011b:46030) Martinón, Antonio A note on measures of nonconvexity. Nonlinear Anal. 72 (2010), no. 6, 3108–3111. (Reviewer: Diana Caponetti), 46B20 (52A05 54B20)

Risultato della ricerca: Other contribution

## Abstract

Eisenfeld and Lakshmikantham [Yokohama Math. J. 24(1976), no.1-2, 133-140; MR0425704 (54$\#$13657)] defined themeasure of nonconvexity $\alpha(C)$ of a subset $C$ of a Banachspace $X$ to be the Hausdorff distance $h(C, {\rm conv} C)$ betweenthe set $C$ and its convex hull. In this note the author, for anonempty bounded subset $C$ of $X$, defines a measure ofnonconvexity $\beta(C)$ as the Hausdorff distance of $C$ to thefamily $bx(X)$ of all nonempty bounded convex subsets of $X$, i.e. $\beta(C)= \inf_{K \in bx(X)}h(C,K ).$ The author studies theproperties of $\beta$. He shows that $\alpha$ and $\beta$ areequivalent, but not equal in the general case.
Lingua originale English Published - 2010