Locally nonconical unit balls in Orlicz spaces

Ryszard Grząślewicz, and Witold Seredyński. "Locally nonconical unit balls in Orlicz spaces." Commentationes Mathematicae 47.1 (2007): null. <http://eudml.org/doc/292203>.

@article{RyszardGrząślewicz2007,
abstract = {The aim of this paper is to investigate the local nonconicality of unit ball in Orlicz spaces, endowed with the Luxemburg norm. A closed convex set $Q$ in a locally convex topological Hausdorff space $X$ is called locally nonconical $(LNC)$, if for every $x, y \in Q$ there exists an open neighbourhood $U$ of $x$ such that $(U\cap Q) + (y - x)/2 \subset Q$. The following theorem is established: An Orlicz space $L^\varphi (\mu )$ has an $LNC$ unit ball if and only if either $L^\varphi (\mu )$ is finite dimensional or the measure $\mu $ is atomic with a positive greatest lower bound and $\varphi $ satisfies the condition $\delta _r^0(\mu )$ and is strictly convex on the interval $[0, b]$, or $c(\varphi ) = +\infty $ and $\varphi $ satisfies the condition $\Delta _2 (\mu )$ and is strictly convex on $\mathbb \{R\}$. A similar result is obtained for the space $E^\varphi (\mu )$.},
author = {Ryszard Grząślewicz, Witold Seredyński},
journal = {Commentationes Mathematicae},
keywords = {stable convex set},
language = {eng},
number = {1},
pages = {null},
title = {Locally nonconical unit balls in Orlicz spaces},
url = {http://eudml.org/doc/292203},
volume = {47},
year = {2007},
}

TY - JOUR
AU - Ryszard Grząślewicz
AU - Witold Seredyński
TI - Locally nonconical unit balls in Orlicz spaces
JO - Commentationes Mathematicae
PY - 2007
VL - 47
IS - 1
SP - null
AB - The aim of this paper is to investigate the local nonconicality of unit ball in Orlicz spaces, endowed with the Luxemburg norm. A closed convex set $Q$ in a locally convex topological Hausdorff space $X$ is called locally nonconical $(LNC)$, if for every $x, y \in Q$ there exists an open neighbourhood $U$ of $x$ such that $(U\cap Q) + (y - x)/2 \subset Q$. The following theorem is established: An Orlicz space $L^\varphi (\mu )$ has an $LNC$ unit ball if and only if either $L^\varphi (\mu )$ is finite dimensional or the measure $\mu $ is atomic with a positive greatest lower bound and $\varphi $ satisfies the condition $\delta _r^0(\mu )$ and is strictly convex on the interval $[0, b]$, or $c(\varphi ) = +\infty $ and $\varphi $ satisfies the condition $\Delta _2 (\mu )$ and is strictly convex on $\mathbb {R}$. A similar result is obtained for the space $E^\varphi (\mu )$.
LA - eng
KW - stable convex set
UR - http://eudml.org/doc/292203
ER -