Abstract
We discuss some open questions on unique determination of convex bodies.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Aleksandrov, A.D.: On the theory of mixed volumes of convex bodies, Part II, New inequalities between mixed volumes and their applications. Mat. Sbornik (N. S.) 2(44), 1205–1238 (1937)
Barker, J.A., Larman, D.G.: Determination of convex bodies by certain sets of sectional volumes. Disc. Math. 241, 79–96 (2001)
Bianchi, G., Gruber, P.M.: Characterization of ellipsoids. Arch. Math. (Basel) 49, 344–350 (1987)
Bonnesen, T.: Om Minkowskis uligheder for konvekse legemer. Mat. Tidsskr. B 74–80 (1926)
Falconer, K.J.: Applications of a result of spherical integration to the theory of convex sets. Am. Math. Mon. 90(10), 690–695 (1983)
Falconer, K.J.: X-ray problems for point sources. Proc. London Math. Soc. 46, 241–262 (1983)
Firey, W.J.: Convex bodies of constant outer p-measure. Mathematika 17, 21-27 (1970)
Gardner, R.J.: Geometric Tomography, 2nd edn. Encyclopedia of Mathematics and its Applications, vol. 58. Cambridge University Press, New York (2006)
Groemer, H.: Geometric Applications of Fourier Series and Spherical Harmonics. Cambridge University Press, New York (1996)
Gardner, R.J., Ryabogin, D., Yaskin, V., Zvavitch, A.: A problem of Klee on inner section functions of convex bodies. J. Diff. Geom., to appear
Golubyatnikov, V.P.: Uniqueness questions in reconstruction of multidimensional objects from tomography type projection data. In: Inverse and Ill-Posed Problems Series. Utrecht-Boston-Köln-Tokyo (2000)
Goodey, P., Schneider, R., Weil, W.: On the determination of convex bodies by projection functions. Bull. London Math. Soc. 29, 82–88 (1997)
Howard, R.: Convex bodies of constant width and constant brightness. Adv. Math. 204, 241261 (2006)
Howard, R., Hug, D.: Smooth convex bodies with proportional projection functions. Israel J. Math. 159, 317–341 (2007)
Howard, R., Hug, D.: Nakajima’s problem: convex bodies of constant width and constant brightness. Mathematika 54, 15–24 (2007)
Howard, R., Nazarov, F., Ryabogin, D., Zvavitch, A.: Determining starlike bodies by the perimeters of their central sections. Preprint
Hug, D.: Nakajima’s problem for general convex bodies. Proc. Amer. Math. Soc. 137, 255–263 (2009)
Klee, V.: Is a body spherical if its HA-measurements are constant? Am. Math. Mon. 76, 539-542 (1969)
Klee, V.: Shapes of the future. Am. Sci. 59, 84-91 (1971)
Koldobsky, A.: Fourier Analysis in Convex Geometry. American Mathematical Society, Providence, RI (2005)
Koldobsky, A., Shane, C.: The determination of convex bodies from derivatives of section functions. Arch. Math. 88, 279–288 (2007)
Koldobsky, A., Yaskin, V.: The interface between convex geometry and harmonic analysis. CBMS Regional Conference Series, vol. 108. American Mathematical Society, Providence, RI (2008)
Makai, E., Martini, H.: On maximal k-sections and related common transversals of convex bodies. Canad. Math. Bull. 47, 246–256 (2004)
Makai, E., Martini, H.: Centrally symmetric convex bodies and sections having maximal quermassintegrals. Studia Sci. Math. Hungar. 49, 189–199 (2012)
Makai, E., Martini, H., Ódor, T.: Maximal sections and centrally symmetric bodies. Mathematika 47, 19–30 (2000)
Nakajima, S.: Eine charakteristische Eigenschaft der Kugel. Jber. Deutsche Math.-Verein 35, 298-300 (1926)
Nazarov, F., Ryabogin, D., Zvavitch, A.: Non-uniqueness of convex bodies with prescribed volumes of sections and projections, accepted to Mathematika
Rusu, A.: Determining starlike bodies by their curvature integrals. Ph.D. Thesis, University of South Carolina (2008)
Ryabogin, D., Yaskin, V.: On counterexamples in questions of unique determination of convex bodies. Proc. Amer. Math. Soc., to appear
Santaló, L.A.: Two characteristic properties of circles on a spherical surface (Spanish). Math. Notae 11, 73-78 (1951)
Schneider, R.: Functional equations connected with rotations and their geometric applications. Enseign. Math. 16, 297–305 (1970)
Schneider, R.: Convex bodies with congruent sections. Bull. London Math. Soc. 12, 52-54 (1980)
Schneider, R.: Convex Bodies: The Brunn-Minkowski Theory. Cambridge University Press, Cambridge (1993)
Xiong, G., Ma, Y.-W., Cheung, W.-S.: Determination of convex bodies from Γ-section functions. J. Shanghai Univ. 12(3), 200–203 (2008)
Yaskin, V.: Unique determination of convex polytopes by non-central sections. Math. Ann. 349, 647–655 (2011)
Yaskin, V.: On perimeters of sections of convex polytopes. J. Math. Anal. Appl. 371, 447–453 (2010)
Acknowledgments
First author supported in part by U.S. National Science Foundation Grants DMS-0652684 and DMS-1101636. Second author supported in part by NSERC. Third author supported in part by U.S. National Science Foundation Grant DMS-1101636.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer Science+Business Media, LLC
About this paper
Cite this paper
Ryabogin, D., Yaskin, V., Zvavitch, A. (2012). Harmonic Analysis and Uniqueness Questions in Convex Geometry. In: Bilyk, D., De Carli, L., Petukhov, A., Stokolos, A., Wick, B. (eds) Recent Advances in Harmonic Analysis and Applications. Springer Proceedings in Mathematics & Statistics, vol 25. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4565-4_26
Download citation
DOI: https://doi.org/10.1007/978-1-4614-4565-4_26
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-4564-7
Online ISBN: 978-1-4614-4565-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)