Abstract
Barker and Larman asked the following. Let \({K' \subset {\mathbb{R}}^d}\) be a convex body, whose interior contains a given convex body \({K \subset {\mathbb{R}}^d}\), and let, for all supporting hyperplanes H of K, the (d − 1)-volumes of the intersections \({K' \cap H}\) be given. Is K′ then uniquely determined? Yaskin and Zhang asked the analogous Question when, for all supporting hyperplanes H of K, the d-volumes of the “caps” cut off from K′ by H are given. We give local positive answers to both of these questions, for small C 2-perturbations of K, provided the boundary of K is C 2+ . In both cases, (d − 1)-volumes or d-volumes can be replaced by k-dimensional quermassintegrals for \({1 \le k \le d-1}\) or for \({1 \le k \le d}\), respectively. Moreover, in the first case we can admit, rather than hyperplane sections, sections by l-dimensional affine planes, where \({1 \le k \le l \le d-1}\). In fact, here not all l-dimensional affine subspaces are needed, but only a small subset of them (actually, a (d − 1)-manifold), for unique local determination of K′.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
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)
Barker J.A., Larman D.G.: Determination of convex bodies by certain sets of sectional volumes. Discrete Math., 241, 79–96 (2001)
Falconer K.: X-ray problems for point sources Proc. London Math. Soc.(3) 46, 241–262 (1983)
Gardner R.J.: Symmetrals and X-rays of planar convex bodies. Arch. Math. (Basel), 41, 183–189 (1983)
R. J. Gardner, Geometric Tomography, second edition, Encyclopedia of Math. and its Appl. 58, Cambridge Univ. Press (Cambridge, 2006).
V. Gorkavyy and D. Kalinin, Barker–Larman problem in the hyperbolic plane, Results Math., 68, 519–525 (2015)
H. Groemer, Geometric Applications of Fourier Series and Spherical Harmonics, Encyclopedia of Math. and its Appl. 61, Cambridge Univ. Press (Cambridge, 1996).
Hammer P.C.: Diameters of convex bodies. Proc. Amer. Math. Soc., 5, 304–306 (1954)
Jerónimo-Castro J., McAllister T.B.: Two characterizations of ellipsoidal cones. J. Convex Anal., 20, 1181–1187 (2013)
Larman D.G.: A note on the false centre theorem. Mathematika, 21, 216–227 (1974)
Larman D.G., Morales-Amaya E.: On the false pole problem. Monatsh Math., 151, 271–286 (2007)
Makai E. Jr., Martini H.: Centrally symmetric convex bodies and sections having maximal quermassintegrals. Studia Sci. Math. Hungar., 49, 189–199 (2012)
Makai E. Jr., Martini H., Ódor T.: Maximal sections and centrally symmetric convex bodies. Mathematika, 47, 17–30 (2000)
Montejano L., Morales-Amaya E.: Variations of classic characterizations of ellipsoids and a short proof of the false centre theorem. Mathematika, 54, 35–40 (2007)
D. Ryabogin, V. Yaskin and A. Zvavitch, Harmonic analysis and uniqueness questions in convex geometry, in: Recent Advances in Harmonic Analysis and Applications, (eds. D. Bilyk et al.), Springer Proc. Math. Stat. 25, Springer (New York, 2013), pp. 327–337.
Santaló L.A.: Two characteristic properties of circles on a spherical surface (Spanish). Math. Notae, 11, 73–78 (1951)
R. Schneider, Convex Bodies: the Brunn–Minkowski Theory, second expanded edition, Encyclopedia of Math. and its Appl., 151, Cambridge Univ. Press (Cambridge, 2014).
Soltan V.: Convex hypersurfaces with hyperplanar intersections of their homothetic copies. J. Convex Anal., 22, 145–159 (2015)
Xiong G., Ma Y.-W., Cheung W.-S.: Determination of convex bodies from \({\Gamma}\)-section functions. J. Shanghai Univ., 12, 200–203 (2008)
Yaskin V.: Unique determination of convex polytopes by non-central sections. Math. Ann., 349, 647–655 (2011)
V. Yaskin and N. Zhang, Non-central sections of convex bodies, http://arxiv.org/abs/1509.08174.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Makai, E., Martini, H. Unique local determination of convex bodies. Acta Math. Hungar. 150, 176–193 (2016). https://doi.org/10.1007/s10474-016-0640-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10474-016-0640-z
Key words and phrases
- questions of Barker–Larman and Yaskin–Zhang
- unique determination of convex bodies
- quermassintegral
- radial function
- star-shaped set
- characterizations of symmetry