On bodies with congruent sections by cones or non-central planes

Zhang, N.

doi:10.1090/tran/7395

On bodies with congruent sections by cones or non-central planes
HTML articles powered by AMS MathViewer

by N. Zhang PDF

Trans. Amer. Math. Soc. 370 (2018), 8739-8756 Request permission

Abstract:

Let $K$ and $L$ be two convex bodies in $\mathbb {R}^3$, such that their sections by cones $\{x\in \mathbb {R}^3:x\cdot \xi =t|x|\}$ or non-central planes with a fixed distance from the origin are directly congruent. We prove that if their boundaries are of class $C^2$, then $K$ and $L$ coincide.

References

M. Angeles Alfonseca, Michelle Cordier, and Dmitry Ryabogin, On bodies with directly congruent projections and sections, Israel J. Math. 215 (2016), no. 2, 765–799. MR 3552295, DOI 10.1007/s11856-016-1394-6
Richard J. Gardner, Geometric tomography, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 58, Cambridge University Press, New York, 2006. MR 2251886, DOI 10.1017/CBO9781107341029
V. Golubyatnikov, Uniqueness questions in reconstruction of multidimensional objects from tomography type projection data, Inverse and Ill-posed problems series, Utrecht-Boston-Koln-Tokyo, 2000.
Alfred Gray, Elsa Abbena, and Simon Salamon, Modern differential geometry of curves and surfaces with Mathematica$^\circledR$, 3rd ed., Studies in Advanced Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2006. MR 2253203
Serge Lang, Introduction to differentiable manifolds, 2nd ed., Universitext, Springer-Verlag, New York, 2002. MR 1931083
Sergii Myroshnychenko and Dmitry Ryabogin, On polytopes with congruent projections or sections, Adv. Math. 325 (2018), 482–504. MR 3742596, DOI 10.1016/j.aim.2017.12.003
Dmitry Ryabogin, On the continual Rubik’s cube, Adv. Math. 231 (2012), no. 6, 3429–3444. MR 2980504, DOI 10.1016/j.aim.2012.09.010
Dmitry Ryabogin, A lemma of Nakajima and Süss on convex bodies, Amer. Math. Monthly 122 (2015), no. 9, 890–892. MR 3418213, DOI 10.4169/amer.math.monthly.122.9.890
J. Sacco, Geometric tomography via conic sections, MSc Thesis, University of Alberta, 2011.
Rolf Schneider, Functions on a sphere with vanishing integrals over certain subspheres, J. Math. Anal. Appl. 26 (1969), 381–384. MR 237723, DOI 10.1016/0022-247X(69)90160-7