TY - JOUR

T1 - On weighted covering numbers and the Levi-Hadwiger conjecture

AU - Artstein-Avidan, Shiri

AU - Slomka, Boaz

N1 - Publisher Copyright:
© 2015, Hebrew University of Jerusalem.
Copyright:
Copyright 2015 Elsevier B.V., All rights reserved.

PY - 2015/9/1

Y1 - 2015/9/1

N2 - We define new natural variants of the notions of weighted covering and separation numbers and discuss them in detail. We prove a strong duality relation between weighted covering and separation numbers and prove a few relations between the classical and weighted covering numbers, some of which hold true without convexity assumptions and for general metric spaces. As a consequence, together with some volume bounds that we discuss, we provide a bound for the famous Levi-Hadwiger problem concerning covering a convex body by homothetic slightly smaller copies of itself, in the case of centrally symmetric convex bodies, which is qualitatively the same as the best currently known bound. We also introduce the weighted notion of the Levi-Hadwiger covering problem, and settle the centrally-symmetric case, thus also confirm the equivalent fractional illumination conjecture [19, Conjecture 7] in the case of centrally symmetric convex bodies (including the characterization of the equality case, which was unknown so far).

AB - We define new natural variants of the notions of weighted covering and separation numbers and discuss them in detail. We prove a strong duality relation between weighted covering and separation numbers and prove a few relations between the classical and weighted covering numbers, some of which hold true without convexity assumptions and for general metric spaces. As a consequence, together with some volume bounds that we discuss, we provide a bound for the famous Levi-Hadwiger problem concerning covering a convex body by homothetic slightly smaller copies of itself, in the case of centrally symmetric convex bodies, which is qualitatively the same as the best currently known bound. We also introduce the weighted notion of the Levi-Hadwiger covering problem, and settle the centrally-symmetric case, thus also confirm the equivalent fractional illumination conjecture [19, Conjecture 7] in the case of centrally symmetric convex bodies (including the characterization of the equality case, which was unknown so far).

UR - http://www.scopus.com/inward/record.url?scp=84945948122&partnerID=8YFLogxK

U2 - 10.1007/s11856-015-1213-5

DO - 10.1007/s11856-015-1213-5

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AN - SCOPUS:84945948122

SN - 0021-2172

VL - 209

SP - 125

EP - 155

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

IS - 1

ER -