Abstract
Among others, we prove that if a convex body \(\mathcal {K}\) and a ball \({\mathcal B}\) have equal constant volumes of caps and equal constant areas of sections with respect to the supporting planes of a sphere, then \({\mathcal K}\equiv {\mathcal B}\).
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Notes
[1, Theorem 1] gives the same conclusion in the plane for disc-isosectioned convex bodies
Although \(\hbar ({\varvec{u}}_{{\varvec{\xi }}},r)=\hbar (-{\varvec{u}}_{{\varvec{\xi }}},-r)\) this parametrization is locally bijective.
Recently Kincses [5] informed the authors in detail that he is very close to finish the construction of two different \(\mathcal D\)-equisectioned convex bodies \(\mathcal K_1\) and \(\mathcal K_2\) in the plane for a disk \(\mathcal D\).
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Acknowledgments
This research was supported by the European Union and co-funded by the European Social Fund under the project “Telemedicine-focused research activities on the field of Mathematics, Informatics and Medical sciences” of project number ‘TÁMOP-4.2.2.A-11/1/KONV-2012-0073”. The authors appreciate János Kincses for discussions of the problems solved in this paper.
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Kurusa, Á., Ódor, T. Characterizations of balls by sections and caps. Beitr Algebra Geom 56, 459–471 (2015). https://doi.org/10.1007/s13366-014-0203-9
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DOI: https://doi.org/10.1007/s13366-014-0203-9