Convex bodies with similar projections
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- by R. J. Gardner and A. Volčič PDF
- Proc. Amer. Math. Soc. 121 (1994), 563-568 Request permission
Abstract:
By examining an example constructed by Petty and McKinney, we show that there are pairs of centered and coaxial bodies of revolution in ${\mathbb {E}^d}, d \geq 3$, whose projections onto each two-dimensional subspace are similar, but which are not themselves even affinely equivalent.References
-
A. D. Alexandrov, On the theory of mixed volumes of convex bodies, II. New inequalities between mixed volumes and their applications, Mat. Sb. 2 (1937), 1205-1238. (Russian)
- G. R. Burton and P. Mani, A characterisation of the ellipsoid in terms of concurrent sections, Comment. Math. Helv. 53 (1978), no. 4, 485–507. MR 511842, DOI 10.1007/BF02566093
- T. Bonnesen and W. Fenchel, Theory of convex bodies, BCS Associates, Moscow, ID, 1987. Translated from the German and edited by L. Boron, C. Christenson and B. Smith. MR 920366
- Ruel V. Churchill, Complex variables and applications, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1960. 2nd ed. MR 0112948
- P. Funk, Über Flächen mit lauter geschlossenen geodätischen Linien, Math. Ann. 74 (1913), no. 2, 278–300 (German). MR 1511763, DOI 10.1007/BF01456044
- V. P. Golubyatnikov, On the unique determination of visible bodies from their projections, Sibirsk. Mat. Zh. 29 (1988), no. 5, 92–96, 238 (Russian); English transl., Siberian Math. J. 29 (1988), no. 5, 761–764 (1989). MR 971230, DOI 10.1007/BF00970269 —, On unique recoverability of convex and visible compacta from their projections, Math. USSR Sb. 73 (1991), 1-10.
- H. Hadwiger, Seitenrisse konvexer Körper und Homothetie, Elem. Math. 18 (1963), 97–98 (German). MR 155232 S. Nakajima, Eine Kennzeichnung homothetische Eiflächen, Tôhoku Math. J. 35 (1932), 285-286.
- C. M. Petty and James R. McKinney, Convex bodies with circumscribing boxes of constant volume, Portugal. Math. 44 (1987), no. 4, 447–455. MR 952791
- C. A. Rogers, Sections and projections of convex bodies, Portugal. Math. 24 (1965), 99–103. MR 198344 W. Süss, Zusammensetzung von Eikörpern und homothetische Eiflächen, Tôhoku Math. J. 35 (1932), 47-50.
Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 121 (1994), 563-568
- MSC: Primary 52A20
- DOI: https://doi.org/10.1090/S0002-9939-1994-1185262-5
- MathSciNet review: 1185262