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DETERMINATION OF A CONVEX BODY FROM MINKOWSKI SUMS OF ITS PROJECTIONS

Published online by Cambridge University Press:  01 October 2004

MARKUS KIDERLEN
Affiliation:
Mathematisches Institut II, University of Karlsruhe, D–76128 Karlsruhe, Germany, kiderlen@math.uni-karlsruhe.de
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Abstract

For a convex body $K$ in $\R^d$ and $1\le k\le d-1$, 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 $k\,{\geq}\, d/2$, $k\,{=}\,3$ for all $d$, and if $k=2$, $d\ne 14$.

It is shown that $P_{2k}(K)$ 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 $P_k$, $1\le k\le d-2$, it is shown that its closure (in the Hausdorff-metric) does not contain any polytopes other than singletons.

Type
Notes and Papers
Copyright
The London Mathematical Society 2004

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