Abstract
One of the most important problems in Geometric Tomography is to establish properties of a given convex body if we know some properties over its sections or its projections. There are many interesting and deep results that provide characterizations of the sphere and the ellipsoid in terms of the properties of its sections or projections. Another kind of characterization of the ellipsoid is when we consider properties of the support cones. However, in almost all the known characterizations, we have only a convex body and the sections, projections, or support cones are considered for this given body. In this article we prove some results that characterize the Euclidean ball or the ellipsoid when the sections or projections are taken for a pair of nested convex bodies, i.e., two convex bodies K, L such that \(L\subset \) int K. We impose some relations between the corresponding sections or projections and some apparently new characterizations of the ball or the ellipsoid appear. We also deal with properties of support cones or point source shadow boundaries when the apexes are taken in the boundary of K.
Similar content being viewed by others
References
Aitchison, P.W., Petty, C.M., Rogers, C.A.: A convex body with a false centre is an ellipsoid. Mathematika 18, 50–59 (1971)
Bianchi, G., Gruber, P.M.: Characterizations of ellipsoids. Arch. Math. (Basel) 49(4), 344–350 (1987)
Blaschke, W.: Kreis und Kugel. Göschen, Leipzig (1916)
Burton, G.R.: Congruent sections of a convex body. Pacific J. Math. 81, 303–319 (1979)
Cieślak, W., Miernowski, A., Mozgawa, W.: Isoptics of a closed strictly convex curve. Lect. Not. Math. 1481, 28–35 (1991)
Green, J.W.: Sets subtending a constant angle on a circle. Duke Math. J. 17, 263–267 (1950)
Groemer, H.: Geometric Applications of Fourier Series and Spherical Harmonics. Cambridge Univ. Press, Cambridge (1996)
Gruber, P.M.: Convex billiards. Geometriae Dedicata 33, 205–226 (1990)
Gruber, P.M.: Only ellipsoids have caustics. Math. Ann. 303, 185–194 (1995)
Gruber, P.M., Ódor, T.: Ellipsoids are the most symmetric convex bodies. Arch. Math. (Basel) 73, 394–400 (1999)
Hadwiger, H.: Vollstdndig stetige Umwendung ebener Eibereiche im Raum. Studies in mathematical analysis and related topics, Stanford University Press, Stanford, pp. 128–131 (1962)
Hammer, P.C.: Diameters of convex bodies. Proc. Am. Math. Soc. 5, 304–306 (1954)
Jerónimo-Castro, J., McAllister, T.B.: Two characterizations of ellipsoidal cones. J. Convex Anal. 20, 1181–1187 (2013)
Jerónimo-Castro, J., Montejano, L., Morales-Amaya, E.: Shaken Rogers’s theorem for homothetic sections. Can. Math. Bull. 52(3), 403–406 (2009)
Larman, D.G., Montejano, L., Morales-Amaya, E.: Characterization of ellipsoids by means of parallel translated sections. Mathematika 56(2), 363–378 (2010)
Mani, P.: Fields of planar bodies tangent to spheres. Monatsh. Math. 74, 145–149 (1970)
Marchaud, A.: Un théoreme sur les corps convexes. Ann. Scient. École Norm. Supér. 76, 283–304 (1959)
Matousek, J.: Using the Borsuk-Ulam Theorem, Lectures on Topological Methods in Combinatorics and Geometry. Universitext, Springer, Berlin (2003)
Montejano, L.: Convex bodies with homothetic sections. Bull. Lond. Math. Soc. 23, 381–386 (1991)
Montejano, L., Morales-Amaya, E.: Variations of classic characterizations of ellipsoids and a short proof of the false centre theorem. Mathematika 54(1–2), 35–40 (2007)
Myroshnychenko, S.: On recognizing shapes of polytopes from their shadows. Discrete Comput. Geom. 62, 856–864 (2019)
Olovjanishnikov, S.P.: On a characterization of the ellipsoid. Ucen. Zap. Leningrad. State Univ. Ser. Mat 83, 114–128 (1941)
Rogers, C.A.: Sections and projections of convex bodies. Port. Math. 24, 99–103 (1965)
Schneider, R.: Convex bodies with congruent sections. Bull. Lond. Math. Soc. 12, 52–54 (1980)
Schütt, C., Werner, E.: The convex floating body. Math. Scand. 66, 275–290 (1990)
Valentine, F.A.: Convex Sets, McGraw-Hill Series in Higher Mathematics. McGraw-Hill, New York (1964)
Werner, E., Ye, D.: On the homothety conjecture. Indiana Univ. Math. J. 60, 1–20 (2011)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
González-García, I., Jerónimo-Castro, J., Morales-Amaya, E. et al. Sections and projections of nested convex bodies. Aequat. Math. 96, 885–900 (2022). https://doi.org/10.1007/s00010-022-00884-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00010-022-00884-4