We consider the class of convex bodies in R(n) with prescribed projection (n - 1)-volumes along finitely many fixed directions. We prove that in such a class there exists a unique body (up to translation) with maximum n-volume. The maximizer is a centrally symmetric polytope and the normal vectors to its facets depend only on the assigned directions. Conditions for the existence of bodies with minimum n-volume in the class defined above are given. Each minimizer is a polytope, and an upper bound for the number of its facets is established.
Campi, S., Colesanti, A., Gronchi, P. (1995). Convex bodies with extremal volumes having prescribed brightness in finitely many directions. GEOMETRIAE DEDICATA, 57(2), 121-133 [10.1007/BF01264932].
Convex bodies with extremal volumes having prescribed brightness in finitely many directions
Campi S.;
1995-01-01
Abstract
We consider the class of convex bodies in R(n) with prescribed projection (n - 1)-volumes along finitely many fixed directions. We prove that in such a class there exists a unique body (up to translation) with maximum n-volume. The maximizer is a centrally symmetric polytope and the normal vectors to its facets depend only on the assigned directions. Conditions for the existence of bodies with minimum n-volume in the class defined above are given. Each minimizer is a polytope, and an upper bound for the number of its facets is established.
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https://hdl.handle.net/11365/32050
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