Abstract
In this paper, we prove that any polygon P in R 2 containing a fixed smooth, strictly convex and origin-symmetric body Γ whose boundary is real analytic in its interior, can be determined by its Γ-section functions among the polygons.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Hammer P C. Problem 2 [C]//Proceedings of the Symposium on Pure Mathematics, vol. VII: Convexity, Providence, Rhode Island. Providence: American Mathematical Society, 1963: 498–499.
Gardner R J. Geometric Tomography [M]. New York: Cambridge University Press, 1995.
Gardner R J. X-rays of polygons [J]. Discrete and Computational Geometry, 1992, 7: 218–293.
Gardner R J, Gritzmann P. Successive determination and verification of polytopes by their X-rays [J]. Journal of the London Mathematical Society, 1994, 50(2): 375–391.
Gardner R J, Soranzo A, Volčcičc A. On the determination of star convex bodies by section functions [J]. Discrete and Computational Geometry, 1999, 21(11): 69–85.
Lam D, Solomon D C. Reconstructing convex polygons in the plane from one directed X-ray [J]. Discrete and Computational Geometry, 2001, 26(1): 256–279.
Kincses J. On the determination of a convex set from its angle function [J]. Discrete and Computational Geometry, 2003, 30(2): 287–297.
Volčič A, Zamifirescu T. Ghosts are scare [J]. Journal of the London Mathematical Society, 1989, 40(2): 171–178.
Barker J A, Larman D G. Determination of convex bodies by certain sets of sectional volumes [J]. Discrete Mathematics, 2001, 241(1–3): 79–96.
Kornfeld I P, Fomin S V, Sinai Y G. Ergodic Theory (Grundlehren der Mathematischen Wissenschaften 245) [M]. [S.l.]: Springer, 1982.
Gruber P M. Baire categories in convexity [M]// Gruber P M, Wills J M. Handbook of Convex Geometry. North-Holland, Amsterdam: [s.n.], 1993: 1327–1346.
Ren Delin. An Introduction to Integral Geometry [M]. Shanghai: Shanghai Scientific and Technical Publishers, 1988 (in Chinese).
Author information
Authors and Affiliations
Corresponding author
Additional information
Project supported by the Youth Science Foundation of Shanghai Municipal Commission of Education (Grant No.214511), and in part by the Research Grants Council of the Hong Kong SAR, China (Grant No.HKU7016/07P)
About this article
Cite this article
Xiong, G., Ma, Yw. & Cheung, Ws. Determination of convex bodies from Γ-section functions. J. Shanghai Univ.(Engl. Ed.) 12, 200–203 (2008). https://doi.org/10.1007/s11741-008-0303-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11741-008-0303-3