Charytanowicz, Katarzyna; Cieślak, Waldemar; Mozgawa, Witold A generalization of the total mean curvature. (English) Zbl 1497.53012 Rend. Semin. Mat. Univ. Padova 146, 223-231 (2021). 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. Reviewer: Esther Sanabria (Valencia) MSC: 53A05 Surfaces in Euclidean and related spaces 52A15 Convex sets in \(3\) dimensions (including convex surfaces) Keywords:support function; total mean curvature; Hopf formula × Cite Format Result Cite Review PDF Full Text: DOI arXiv 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.