Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-24T11:44:18.337Z Has data issue: false hasContentIssue false
  • English
  • Français

When do sections of different dimensions determine a convex body?

Part of: General convexity

Published online by Cambridge University Press:  26 February 2010

Alessandro Soranzo  and
Aljoša Volčič
Alessandro Soranzo
Affiliation:
Dipartimento di Scienze Matematiche, Università degli Studi di Trieste, 34127 Trieste. Italy. E-mail: [email protected]
Aljoša Volčič
Affiliation:
Dipartimento di Scienze Matematiche, Università degli Studi di Trieste, 34127 Trieste, Italy
Get access

Abstract

This paper gives a partial answer to a problem posed by Volčič and shows, in particular, that a three-dimensional convex body K is uniquely determined if p′ and p″ are two points interior to K and the lengths of all the chords of K through p′ and the areas of all sections of K with planes through p″ are known, provided that a specific condition on the positions of p′ and p″ with respect to K is satisfied. The problem will be studied in the more general framework of i-chord functions, and the results will also cover cases where the points p′ and p″ are not interior to K, possibly with one of them at infinity.

MSC classification

Secondary: 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces)
Type
Research Article
Information
Mathematika , Volume 50 , Issue 1-2 , December 2003 , pp. 35 - 55
Copyright
Copyright © University College London 2003

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[BL]Barker, J. A. and Larman, D. G.. Determination of convex bodies by certain sets of sectional volumes. Discrete Math., 241 (2001), 79–96.CrossRefGoogle Scholar
[F1]Falconer, K. J.. X-ray problems for point sources. Proc. London Math. Soc., 46 (1983). 241–262.CrossRefGoogle Scholar
[F2]Falconer, K. J.. Hammer's X-ray problem and the stable manifold theorem. Geom. Dedicata, 14 (1983), 113–126.Google Scholar
[F3]Falconer, K. J.. On the equireciprocal point problem. J. London Math. Soc. (2). 28 (1983). 149–160.CrossRefGoogle Scholar
[Fu]Funk, P.. Über Flächen mit lauter geschlossenen geodätischen Linien. Math. Ann., 74 (1913), 278–300.CrossRefGoogle Scholar
[G]Gardner, R. J.. Geometric Tomography. Cambridge University Press, 1995.Google Scholar
[G1]Gardner, R. J.. Symmetrals and X-rays of planar convex bodies. Archiv Mat., 41 (1983), 183–189.CrossRefGoogle Scholar
[G2]Gardner, R. J.. Chord functions of convex bodies. J. London Math. Soc. (2), 36 (1987), 314–326.CrossRefGoogle Scholar
[G3]Gardner, R. J.. Measure theory and some problems in geometry. Atti Sent. Mat. Fis. Univ. Modena, 39 (1991). 51–72.Google Scholar
[GM]Gardner, R. J. and McMullen, P.. On Hammer's X-ray problem. J. London Math. Soc. (2), 21 (1980), 171–175.CrossRefGoogle Scholar
[GV]Gardner, R. J. and Volčič, A.. Tomography of convex and star bodies. Advances Math., 108 (1994), 367–399.CrossRefGoogle Scholar
[GSV]Gardner, R. J., Soranzo, A. and Volčič, A.. On the determination of star and convex bodies by section functions. J. Disc. Comput. Geom., 21 (1999), 69–85.CrossRefGoogle Scholar
[Gi]Giering, O.. Bestimmung von Eibereichen und Eikörpern durch Steiner-Symmetrisierungen, Sher. Bayer. Akad. Wiss. Münehen, Math.- Nat. Kl. (1962), 225–253.Google Scholar
[H]Hammer, P. C.. Problem 2; in: Proc. Symp. Pure Math., Vol. VII: Convexity. Amer. Math. Soc. 1963.Google Scholar
[K1]Klain, D.. Star valuations and dual mixed volumes. Advances Math., 121 (1996), 80–101.CrossRefGoogle Scholar
[K2]Klain, D.. Invariant valuations on star-shaped sets. Advances Math., 125 (1997), 95–113.CrossRefGoogle Scholar
[Ke]Kelly, J. B.. Power points. Amer. Math. Monthly, 53 (1946), 395–396.Google Scholar
[Ku]Kubota, T.. Einige Probleme über konvex-geschlossene Kurven und Flächen. Tôhoku Math. J., 17 (1920), 351–362.Google Scholar
[L]Lutwak, E.. Dual mixed volumes. Pacific J. Math., 58 (1975), 531–538.CrossRefGoogle Scholar
[LT]Larman, D. G. and Tamvakis, N. K.. A characterization of centrally symmetric convex bodies in Geom. Dedicata, 10 (1981), 161–176.CrossRefGoogle Scholar
[M]Michelacci, G.. On a partial extension of a theorem of Falconer. Ricerche Mat., 37 (1988), 213–220.Google Scholar
[N]Natterer, F.. The Mathematics of Computerized Tomography. Wiley, 1986.Google Scholar
[S]Schneider, R.. Functional equations connected with rotations and their geometric applications, Enseign. Math., 16 (1970), 297–305.Google Scholar
[Sü]Süss, W.. Eibereiche mit ausgezeichneten Punkten; Sehnen-, Inhalts- und Umfangspunkte. Tôhoku Math. J., 25 (1925), 86–98.Google Scholar
[V1]Volčič, A.. A three-point solution to Hammer's X-ray problem. J. London Math. Soc. (2), 34 (1986), 340–359.Google Scholar
[V2]Volčič, A.. Generalized Hammer's X-ray problem. Atti Sem. Mat. Fis. Univ. Modena, (Suppl.) XLVI (1998), 393–400.Google Scholar
[Y]Yanagihara, K.. On a characteristic property of the circle and the sphere. Tôhoku Math. J., 10 (1916), 142–143.Google Scholar
[Z]Zuccheri, L.. Characterization of the circle by equipower points. Archiv Mat., 58 (1992), 199–208.CrossRefGoogle Scholar