Revista Matemática Iberoamericana


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Volume 33, Issue 4, 2017, pp. 1197–1218
DOI: 10.4171/RMI/968

Published online: 2017-11-17

Bounding the integral of powered $i$-th mean curvatures

David Alonso-Gutiérrez[1], María A. Hernández Cifre[2] and Antonio R. Martínez Fernández[3]

(1) Universidad de Zaragoza, Spain
(2) Universidad de Murcia, Spain
(3) Universidad de Murcia, Spain

We get estimates for the integrals of powered $i$-th mean curvatures, $1\leq i\leq n-1$, of compact and convex hypersurfaces, in terms of the quermaß integrals of the corresponding $C^2_+$ convex bodies. These bounds will be obtained as consequences of a most general result for functions defined on a general probability space. From this result, similar estimates for the integrals of any convex transformation of the elementary symmetric functions of the radii of curvature of $C^2_+$ convex bodies will be also proved, both, in terms of the quermaß integrals, and of the roots of their Steiner polynomials. Finally, the radial function is considered, and estimates of the corresponding integrals are obtained in terms of the dual quermaß integrals.

Keywords: Convex hypersurfaces, $C^2_+$ convex bodies, mean curvatures, symmetric functions of radii of curvature, quermaßintegrals, inner and outer radii, roots of Steiner polynomials, radial function, dual quermaßintegrals

Alonso-Gutiérrez David, Hernández Cifre María, Martínez Fernández Antonio: Bounding the integral of powered $i$-th mean curvatures. Rev. Mat. Iberoamericana 33 (2017), 1197-1218. doi: 10.4171/RMI/968