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E. Makai, H. Martini (2012)
Centrally symmetric convex bodies and sections having maximal quermassintegralsarXiv: Metric Geometry
V. Yaskin (2018)
An Extension of Polynomial Integrability to Dual QuermassintegralsInternational Mathematics Research Notices
E. Makai, H. Martini, T. Ódor (2000)
Maximal sections and centrally symmetric bodiesMathematika, 47
Guangxian Zhu (2014)
The logarithmic Minkowski problem for polytopesAdvances in Mathematics, 262
V. Yaskin (2010)
On perimeters of sections of convex polytopesJournal of Mathematical Analysis and Applications, 371
P. Funk (1915)
Über eine geometrische Anwendung der Abelschen IntegralgleichungMathematische Annalen, 77
V. Yaskin, M. Yaskina (2015)
Thick sections of convex bodiesAdv. Appl. Math., 71
H. Groemer (1996)
Geometric Applications of Fourier Series and Spherical Harmonics
R. Schneider (1993)
Convex Bodies: The Brunn–Minkowski Theory: Minkowski addition
KJ Falconer (1983)
690Am. Math. Mon., 90
S Myroshnychenko, D Ryabogin (2018)
On polytopes with congruent projections or sectionsAdv. Math., 325
R. Gardner (1995)
Geometric Tomography
P Funk (1915)
129Math. Ann., 77
S. Myroshnychenko, D. Ryabogin (2016)
On polytopes with congruent projections or sectionsarXiv: Metric Geometry
D. Ryabogin, V. Yaskin (2012)
Detecting symmetry in star bodiesJournal of Mathematical Analysis and Applications, 395
M Stephen, V Yaskin (2017)
Stability results for sections of convex bodiesTrans. Am. Math. Soc., 369
E. Lutwak (1975)
Dual mixed volumesPacific Journal of Mathematics, 58
A. Koldobsky (2005)
Fourier Analysis in Convex Geometry
K. Falconer (1983)
Applications of a Result on Spherical Integration to the Theory of Convex SetsAmerican Mathematical Monthly, 90
E Makai, H Martini (2012)
Centrally symmetric convex bodies and sections having maximal quermassintegralsStudia Sci. Math. Hungar., 49
V. Yaskin (2011)
Unique determination of convex polytopes by non-central sectionsMathematische Annalen, 349
M. Stephen, V. Yaskin (2015)
Stability results for sections of convex bodiesarXiv: Metric Geometry
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R. Schneider (1970)
Über eine Integralgleichung in der Theorie der konvexen KörperMathematische Nachrichten, 44
Let P⊂Rn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$P\subset {\mathbb {R}}^n$$\end{document}(n≥3)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(n\ge 3)$$\end{document} be a convex polytope containing the origin in its interior. Let voln-2(relbd(P∩{tξ+ξ⊥}))\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\text {vol}}_{n-2}({{\text {relbd}}(P\cap \lbrace t\xi + \xi ^\perp \rbrace )})$$\end{document} denote the (n-2)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(n-2)$$\end{document}-dimensional volume of the relative boundary of P∩{tξ+ξ⊥}\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$P\cap \lbrace t\xi + \xi ^\perp \rbrace $$\end{document} for t∈R\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$t\in {\mathbb {R}}$$\end{document}, ξ∈Sn-1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\xi \in S^{n-1}$$\end{document}. We prove the following: if voln-2(relbd(P∩ξ⊥))=maxt∈Rvoln-2(relbd(P∩{tξ+ξ⊥}))\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} {\text {vol}}_{n-2}({{\text {relbd}}(P\cap \xi ^\perp )}) =\max _{t\in {\mathbb {R}}}\,{\text {vol}}_{n-2}({{\text {relbd}}(P\cap \lbrace t\xi + \xi ^\perp \rbrace )}) \end{aligned}$$\end{document}for all ξ∈Sn-1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\xi \in S^{n-1}$$\end{document}, then P is origin-symmetric, i.e., P=-P\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$P=-P$$\end{document}. Our result gives a partial affirmative answer to a conjecture by Makai et al. We also characterize the origin-symmetry of C2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$C^2$$\end{document} star bodies in terms of the dual quermassintegrals of their sections; this can be seen as a dual version of the conjecture of Makai et al.
Discrete & Computational Geometry – Springer Journals
Published: Sep 1, 2022
Keywords: Convex bodies; Convex polytopes; Origin-symmetry; Sections; 52B15; 52A20; 52A38
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