Abstract
Let K and L be two convex bodies in R4, such that their projections onto all 3-dimensional subspaces are directly congruent. We prove that if the set of diameters of the bodies satisfies an additional condition and some projections do not have certain π-symmetries, then K and L coincide up to translation and an orthogonal transformation. We also show that an analogous statement holds for sections of star bodies, and prove the n-dimensional versions of these results.
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A. D. Alexandrov, On the theory of mixed volumes of convex bodies II. New inequalities between mixed volumes and their applications [In Russian], Mat. Sbornik, 2 (1937), 1205–1238.
M. Alfonseca and M. Cordier, Counterexamples related to rotations of shadows of convex bodies, Indiana Univ. Math. J., to appear, http://arxiv.org/abs/1505. 05817.
T. Bonnesen and Fenchel, Theory of Convex Bodies, BCS Associates, Moscow, Idaho, 1987.
R. J. Gardner, Geometric Tomography, second edition, in Encyclopedia of Mathematics and its Applications, 58, Cambridge University Press, Cambridge, 2006.
V. P. Golubyatnikov, Uniqueness Questions in Reconstruction of Multidimensional Objects from Tomography Type Projection Data. Inverse and Ill–posed Problems Series. De Gruyter, Utrecht–Boston–Köln–Tokyo, 2000.
H. Hadwiger, Seitenrisse konvexer Körper und Homothetie, Elem.Math. 18 (1963), 97–98.
S. Helgason, Radon Transform, second edition, Birkhäuser, Basel, 1999.
B. Mackey, Convex bodies with SO(2)–congruent projections, Master Thesis, Kent State University, 2012.
J. Milnor, Analytic proofs of the “Hairy ball theorem” and the Brouwer fixed point theorem, Amer. Math. Monthly 85 (1978), 521–524.
R. Palais, On the existence of slices for actions of non–compact Lie groups, Ann. of Math. (2) 73 (1961), 295–323.
C. Radin and L. Sadun, On 2–generator subgroups of SO(3), Trans. Amer. Math. Soc. 351 (1999), 4469–4480.
D. Ryabogin, On the continual Rubik’s cube, Adv. Math. 231 (2012), 3429–3444.
D. Ryabogin, A Lemma of Nakajima and Süss on convex bodies, Amer. Math. Monthly 122 (2015), 890–892.
H. Samelson, A Theorem on differentiable manifolds, Portugaliae Math. 10 (1951), 129–133.
R. Schneider, Convex Bodies: The Brunn–Minkowski Theory, in Encyclopedia of Mathematics and its Applications, 44, Cambridge University Press, Cambridge, 1993.
R. Schneider, Convex bodies with congruent sections, Bull. London Math. Soc. 312 (1980), 52–54.
R. Schneider, Zur optimalen Approximation konvexer Hyperflächen durch Polyeder, Math. Ann. 256 (1981), 289–301.
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Supported in part by U.S. National Science Foundation Grant DMS-1100657.
Supported in part by U.S. National Science Foundation Grants DMS-0652684 and DMS-1101636.
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Alfonseca, M.A., Cordier, M. & Ryabogin, D. On bodies with directly congruent projections and sections. Isr. J. Math. 215, 765–799 (2016). https://doi.org/10.1007/s11856-016-1394-6
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DOI: https://doi.org/10.1007/s11856-016-1394-6