89,413 research outputs found
An isoperimetric inequality on the balls
The normalised volume measure on the unit ball ()
satisfies the following isoperimetric inequality: the boundary measure of a set
of measure is at least , where
.Comment: Published in at http://dx.doi.org/10.1214/07-AIHP121 the Annales de
l'Institut Henri Poincar\'e - Probabilit\'es et Statistiques
(http://www.imstat.org/aihp/) by the Institute of Mathematical Statistics
(http://www.imstat.org
Polya's inequalities, global uniform integrability and the size of plurisubharmonic lemniscates
First we prove a new inequality comparing uniformly the relative volume of a
Borel subset with respect to any given complex euclidean ball \B \sub \C^n
with its relative logarithmic capacity in \C^n with respect to the same ball
\B.
An analoguous comparison inequality for Borel subsets of euclidean balls of
any generic real subspace of \C^n is also proved.
Then we give several interesting applications of these inequalities.
First we obtain sharp uniform estimates on the relative size of \psh
lemniscates associated to the Lelong class of \psh functions of logarithmic
singularities at infinity on \C^n as well as the Cegrell class of
\psh functions of bounded Monge-Amp\`ere mass on a hyperconvex domain \W
\Sub \C^n.
Then we also deduce new results on the global behaviour of both the Lelong
class and the Cegrell class of \psh functions.Comment: 25 page
On contact numbers of totally separable unit sphere packings
Contact numbers are natural extensions of kissing numbers. In this paper we
give estimates for the number of contacts in a totally separable packing of n
unit balls in Euclidean d-space for all n>1 and d>1.Comment: 11 page
Carleson measures and uniformly discrete sequences in strongly pseudoconvex domains
We characterize using the Bergman kernel Carleson measures of Bergman spaces
in strongly pseudoconvex bounded domains in several complex variables,
generalizing to this setting theorems proved by Duren and Weir for the unit
ball. We also show that uniformly discrete (with respect to the Kobayashi
distance) sequences give examples of Carleson measures, and we compute the
speed of escape to the boundary of uniformly discrete sequences in strongly
pseudoconvex domains, generalizing results obtained in the unit ball by
Jevti\'c, Massaneda and Thomas, by Duren and Weir, and by MacCluer.Comment: 17 page
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