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Matheron's Conjecture for the Covariogram Problem
Author(s) -
Bianchi Gabriele
Publication year - 2005
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610704006039
Subject(s) - conjecture , regular polygon , convex body , boundary (topology) , mathematics , combinatorics , pure mathematics , convex hull , mathematical analysis , geometry
The covariogram of a convex body K provides the volumes of the intersections of K with all its possible translates. Matheron conjectured in 1986 that this information determines K among all convex bodies, up to certain known ambiguities. It is proved that this is the case if K ⊂ R 2 is not C 1 , or it is not strictly convex, or its boundary contains two arbitrarily small C 2 open portions ‘on opposite sides’. Examples are also constructed that show that this conjecture is false in R n for any n ⩾ 4.