Submitted Paper

Inserted: 16 dec 2013
Last Updated: 16 dec 2013

Year: 2013

Abstract:

Let K ⊂ RN be any convex body containing the origin. A measurable set G ⊂ RN with finite and positive Lebesgue measure is said to be K-dense if, for any fixed r > 0, the measure of G∩(x+rK) is constant when x varies on the boundary of G (here, x + rK denotes a translation of a dilation of K). In 6, we proved for the case in which N = 2 that if G is K-dense, then both G and K must be homothetic to the same ellipse. Here, we completely characterize K-dense sets in RN : if G is K-dense, then both G and K must be homothetic to the same ellipsoid. Our proof, by building upon results obtained in 6, relies on an asymptotic formula for the measure of G ∩ (x + rK) for large values of the parameter r and a classical characterization of ellipsoids due to C.M. Petty 8.

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• Kdense