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COVARIOGRAM OF NON-CONVEX SETS

Part of:Harmonic analysis in several variablesGeneral convexityGeometric probability and stochastic geometry

Published online by Cambridge University Press:  21 June 2010

Carlo Benassi,
Gabriele Bianchi and
Giuliana D’Ercole
Carlo Benassi
Affiliation:
Dipartimento di Matematica, Università di Modena e Reggio Emilia, via Campi 213/B, Modena, I-41100, Italy (email: benassi.carlo@unimore.it)
Gabriele Bianchi
Affiliation:
Dipartimento di Matematica, Università di Firenze, Viale Morgagni 67/A, Firenze, I-50134, Italy (email: gabriele.bianchi@unifi.it)
Giuliana D’Ercole
Affiliation:
Dipartimento di Matematica, Università di Modena e Reggio Emilia, via Campi 213/B, Modena, I-41100, Italy (email: giulianadercole@libero.it)

Abstract

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The covariogram of a compact set A⊂ℝn is the function that to each x∈ℝn associates the volume of A∩(A+x). Recently it has been proved that the covariogram determines any planar convex body, in the class of all convex bodies. We extend the class of sets in which a planar convex body is determined by its covariogram. Moreover, we prove that there is no pair of non-congruent planar polyominoes consisting of less than nine points that have equal discrete covariograms.

MSC classification

Secondary:60D05: Geometric probability and stochastic geometry52A10: Convex sets in $2$ dimensions (including convex curves)52A22: Random convex sets and integral geometry52A38: Length, area, volume42B10: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type

Information

Type
Research Article
Information
Mathematika , Volume 56 , Issue 2 , July 2010, pp. 267 - 284
Copyright
Copyright © University College London 2010

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