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Determining convex polygons from their covariograms

Part of: Geometric probability and stochastic geometry General convexity

Published online by Cambridge University Press:  01 July 2016

Gabriele Bianchi
Gabriele Bianchi*
Affiliation:
Università degli Studi di Ferrara
*
∗ Current address: Dipartimento di Ingegneria Agraria e Forestale, Università di Firenze, Piazzale delle Cascine 15, I-50144 Firenze, Italy. Email address: gabriele.bianchi@unifi.it
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Abstract

Knowing the (geometric) covariogram of a convex body is equivalent to knowing, for each direction u, the distribution of the lengths of the chords of that body which are parallel to u. We prove that the covariogram determines convex polygons, among all convex bodies, up to translation and reflection. This gives a partial answer to a problem posed by Matheron.

Keywords

Convex polygonconvex bodycovariogramphase retrieval

MSC classification

Primary: 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces)
Secondary: 60D05: Geometric probability and stochastic geometry 52A22: Random convex sets and integral geometry 52A38: Length, area, volume
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 34 , Issue 2 , June 2002 , pp. 261 - 266
Copyright
Copyright © Applied Probability Trust 2002 

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