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Determination of a Convex Body from Minkowski Sums of its Projections
Author(s) -
Kiderlen Markus
Publication year - 2004
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/s0024610704005551
Subject(s) - convex body , mathematics , combinatorics , regular polygon , convex polytope , minkowski space , subderivative , hausdorff distance , injective function , convex combination , convex analysis , mathematical analysis , convex hull , geometry , convex optimization
For a convex body K in R d and 1 ⩽ K ⩽ d − 1, let P K ( K ) be the Minkowski sum (average) of all orthogonal projections of K onto k ‐dimensional subspaces of R d . It is Known that the operator P k is injective if k ⩾d/2, k =3 for all d , and if k = 2, d ≠ 14. It is shown that P 2 k ( K ) determines a convex body K among all centrally symmetric convex bodies and P 2k+1 ( K ) determines a convex body K among all bodies of constant width. Corresponding stability results are also given. Furthermore, it is shown that any convex body K is determined by the two sets P k ( K ) and P k′ ( K ) if 1 < k < k ′. Concerning the range of P k , 1 ⩽ k ⩽ d−2, it is shown that its closure (in the Hausdorff‐metric) does not contain any polytopes other than singletons.