- journal article metadata
European Mathematical Society Publishing House
2017-11-20 23:40:01
Revista Matemática Iberoamericana
Rev. Mat. Iberoamericana
RMI
0213-2230
2235-0616
General
10.4171/RMI
http://www.ems-ph.org/doi/10.4171/RMI
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European Mathematical Society Publishing House
Zuerich, Switzerland
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33
2017
4
Bounding the integral of powered $i$-th mean curvatures
David
Alonso-Gutiérrez
Universidad de Zaragoza, Spain
María
Hernández Cifre
Universidad de Murcia, Spain
Antonio
Martínez Fernández
Universidad de Murcia, Spain
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
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.
Convex and discrete geometry
Combinatorics
Differential geometry
1197
1218
10.4171/RMI/968
http://www.ems-ph.org/doi/10.4171/RMI/968
11
17
2017