NDLI: Characterization of ellipses as uniformly dense sets with respect to a family of convex bodies

Content Provider	Springer Nature Link
Author	Marini, Michele Magnanini, Rolando
Copyright Year	2013
Abstract	Let $$K\subset \mathbb R ^N$$ be a convex body containing the origin. A measurable set $$G\subset \mathbb R ^N$$ with positive Lebesgue measure is said to be uniformly $$K$$ -dense if, for any fixed $$r>0$$ , the measure of $$G\cap (x+r K)$$ is constant when $$x$$ varies on the boundary of $$G$$ (here, $$x+r K$$ denotes a translation of a dilation of $$K$$ ). We first prove that $$G$$ must always be strictly convex and at least $$C^{1,1}$$ -regular; also, if $$K$$ is centrally symmetric, $$K$$ must be strictly convex, $$C^{1,1}$$ -regular and such that $$K=G-G$$ up to homotheties; this implies in turn that $$G$$ must be $$C^{2,1}$$ -regular. Then for $$N=2$$ , we prove that $$G$$ is uniformly $$K$$ -dense if and only if $$K$$ and $$G$$ are homothetic to the same ellipse. This result was already proven by Amar et al. in 2008 . However, our proof removes their regularity assumptions on $$K$$ and $$G$$ , and more importantly, it is susceptible to be generalized to higher dimension since, by the use of Minkowski’s inequality and an affine inequality, avoids the delicate computations of the higher-order terms in the Taylor expansion near $$r=0$$ for the measure of $$G\cap (x+r\,K)$$ (needed in 2008).
Ending Page	1395
Page Count	13
Starting Page	1383
File Format	PDF
ISSN	03733114
e-ISSN	16181891
Journal	Annali di Matematica Pura ed Applicata
Issue Number	5
Volume Number	193
Language	English
Publisher	Springer Berlin Heidelberg
Publisher Date	2013-03-24
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Inequalities and extremum problems Convex bodies Affine inequalities Convex sets in $2$ dimensions (including convex curves) Convex sets in $n$ dimensions (including convex hypersurfaces) Mixed volumes and related topics Mathematics Uniformly dense sets
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics

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