Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 7, Tome 280 (2001), pp. 141-145
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N. M. Gulevich; O. N. Gulevich. An estimate for the measure of nonconvexity in the $L^p$-space. Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 7, Tome 280 (2001), pp. 141-145. http://geodesic.mathdoc.fr/item/ZNSL_2001_280_a6/
@article{ZNSL_2001_280_a6,
author = {N. M. Gulevich and O. N. Gulevich},
title = {An estimate for the measure of nonconvexity in the $L^p$-space},
journal = {Zapiski Nauchnykh Seminarov POMI},
pages = {141--145},
year = {2001},
volume = {280},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/ZNSL_2001_280_a6/}
}
TY - JOUR
AU - N. M. Gulevich
AU - O. N. Gulevich
TI - An estimate for the measure of nonconvexity in the $L^p$-space
JO - Zapiski Nauchnykh Seminarov POMI
PY - 2001
SP - 141
EP - 145
VL - 280
UR - http://geodesic.mathdoc.fr/item/ZNSL_2001_280_a6/
LA - ru
ID - ZNSL_2001_280_a6
ER -
%0 Journal Article
%A N. M. Gulevich
%A O. N. Gulevich
%T An estimate for the measure of nonconvexity in the $L^p$-space
%J Zapiski Nauchnykh Seminarov POMI
%D 2001
%P 141-145
%V 280
%U http://geodesic.mathdoc.fr/item/ZNSL_2001_280_a6/
%G ru
%F ZNSL_2001_280_a6
The measure $\alpha(A)$ of nonconvexity for a bounded subset $A$ of a normed linear space $L$ is the Hausdorff distance between $A$ and its convex hull co $A$. It is proved that if $L$ is an $L^p$-space, then $\alpha(A)\le d(A)/2^{t_p}$, where $d(A)$ is the diameter of $A$ and $t_p=\min\{1/p,1-1/p\}$, $1\le p\le\infty$.Furthermore, this estimate is sharp.
