Eisenfeld and Lakshmikantham [Yokohama Math. J. 24 (1976), no.1-2, 133-140; MR0425704 (54$\#$13657)] defined the measure of nonconvexity $\alpha(C)$ of a subset $C$ of a Banach space $X$ to be the Hausdorff distance $h(C, {\rm conv} C)$ between the set $C$ and its convex hull. In this note the author, for a nonempty bounded subset $C$ of $X$, defines a measure of nonconvexity $\beta(C)$ as the Hausdorff distance of $C$ to the family $bx(X)$ of all nonempty bounded convex subsets of $X$, i.e. $\beta(C)= \inf_{K \in bx(X)}h(C,K ).$ The author studies the properties of $\beta$. He shows that $\alpha$ and $\beta$ are equivalent, but not equal in the general case.

CAPONETTI, D. (2010). 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).

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)

Abstract

Eisenfeld and Lakshmikantham [Yokohama Math. J. 24 (1976), no.1-2, 133-140; MR0425704 (54$\#$13657)] defined the measure of nonconvexity $\alpha(C)$ of a subset $C$ of a Banach space $X$ to be the Hausdorff distance $h(C, {\rm conv} C)$ between the set $C$ and its convex hull. In this note the author, for a nonempty bounded subset $C$ of $X$, defines a measure of nonconvexity $\beta(C)$ as the Hausdorff distance of $C$ to the family $bx(X)$ of all nonempty bounded convex subsets of $X$, i.e. $\beta(C)= \inf_{K \in bx(X)}h(C,K ).$ The author studies the properties of $\beta$. He shows that $\alpha$ and $\beta$ are equivalent, but not equal in the general case.
Scheda breve Scheda completa Scheda completa (DC)
CAPONETTI, D. (2010). 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).
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10447/55819
• ND
• ND
• ND