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ON THE RECONSTRUCTION OF BODIES FROM THEIR PROJECTIONS OR SECTIONS

Myroshnychenko, Sergii
http://rave.ohiolink.edu/etdc/view?acc_num=kent1501536418525866

Abstract Details

2017, PHD, Kent State University, College of Arts and Sciences / Department of Mathematical Sciences.
We consider two long-standing open problems of geometric tomography: Problem 1: Let k,d be two fixed integers, 2=k=d-1. Assume that K and L are convex bodies in Euclidean space E^d such that their projections K|H and L|H are congruent for each subspace H, dim¿(H) = k. Is L a translate of K or -K? Problem 2: Let k,d be two fixed integers, 2=k=d-1. Assume that K and L are star bodies in Euclidean space E^d such that their sections KnH and LnH are congruent for each subspace H, dim¿(H) = k. Does it imply that L=K or L=-K? We give the affirmative answers to Problems 1 and 2 in the class of convex polytopes. We also prove an affirmative result related to Problem 1: two classical hedgehogs (a class of possibly singular, self-intersecting and non-convex surfaces) coincide up to a translation and a reflection in the origin, provided that their projections onto any 2-dim subspace are directly congruent and have no direct rigid motion symmetries. This is a consequence of a more general analytic statement about the solutions of a functional equation.
Dmitry Ryabogin (Advisor)
78 p.
Mathematics
convex geometry, geometric tomography

Recommended Citations

Citations

  • Myroshnychenko, S. (2017). ON THE RECONSTRUCTION OF BODIES FROM THEIR PROJECTIONS OR SECTIONS [Doctoral dissertation, Kent State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=kent1501536418525866

    APA Style (7th edition)

  • Myroshnychenko, Sergii. ON THE RECONSTRUCTION OF BODIES FROM THEIR PROJECTIONS OR SECTIONS. 2017. Kent State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=kent1501536418525866.

    MLA Style (8th edition)

  • Myroshnychenko, Sergii. "ON THE RECONSTRUCTION OF BODIES FROM THEIR PROJECTIONS OR SECTIONS." Doctoral dissertation, Kent State University, 2017. http://rave.ohiolink.edu/etdc/view?acc_num=kent1501536418525866

    Chicago Manual of Style (17th edition)

kent1501536418525866
336
© 2017, all rights reserved.
This open access ETD is published by Kent State University and OhioLINK.