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A generalization of the total mean curvature. (English) Zbl 1497.53012

Authors’ abstract: The authors derive a special formula for the total mean curvature of an ovaloid. This formula allows us to extend the notion of the mean curvature to the class of boundaries of strictly convex sets. Moreover, an integral formula is proved.

MSC:

53A05 Surfaces in Euclidean and related spaces
52A15 Convex sets in \(3\) dimensions (including convex surfaces)

References:

[1] R. Alexander, Lipschitzian mappings and total mean curvature of polyhedral sur-faces. I, Trans. Amer. Math. Soc. 288 (1985), no. 2, pp. 661-678. · Zbl 0563.52008
[2] T. Bonnesen -W. Fenchel, Theory of convex bodies, translated from the German and edited by L. Boron, C. Christenson, and B. Smith, BCS Associates, Moscow, ID, 1987. · Zbl 0628.52001
[3] Yu. D. Burago -V. A. Zalgaller, Geometric inequalities, Translated from the Russian by A. B. Sosinskiȋ, Grundlehren der Mathematischen Wissenschaften, 285, Springer Series in Soviet Mathematics, Springer, Berlin, 1988. · Zbl 0633.53002
[4] K. Charytanowicz -W. Cieślak -W. Mozgawa, A new formula for the length of a closed curve, Beitr. Algebra Geom. 61 (2020), no. 3, pp. 465-472. · Zbl 1444.53005
[5] M. Hazewinkel (ed.), Encyclopaedia of mathematics, Vol. 7. Orbit-Rayleigh equation, Translated from the Russian, Kluwer Academic Publishers Group, Dordrecht, 1991. · Zbl 0806.00006
[6] H. Hopf, Differential geometry in the large, Notes taken by P. Lax and J. Gray, With a preface by S. S. Chern, Lecture Notes in Mathematics, 1000, Springer, Berlin, 1983. · Zbl 0526.53002
[7] L. A. Santaló, Integral geometry and geometric probability, Second edition, with a foreword by Mark Kac. Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2004. · Zbl 1116.53050
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