Convex sets with homothetic projections
نویسنده
چکیده
Nonempty sets X1 and X2 in the Euclidean space R n are called homothetic provided X1 = z+λX2 for a suitable point z ∈ R n and a scalar λ 6= 0, not necessarily positive. Extending results of Süss and Hadwiger (proved by them for the case of convex bodies and positive λ), we show that compact (respectively, closed) convex sets K1 and K2 in R n are homothetic provided for any given integer m, 2 ≤ m ≤ n − 1 (respectively, 3 ≤ m ≤ n − 1), the orthogonal projections of K1 and K2 on every m-dimensional plane of R n are homothetic, where homothety ratio may depend on the projection plane. The proof uses a refined version of Straszewicz’s theorem on exposed points of compact convex sets. AMS Subject Classification: 52A20.
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