Abstract
LetF be a closed convex hypersurface in Euclideand-space with almost constantq-th mean curvatureH q (q=1, ...,d−1). The deviation ofF from a suitable sphere is estimated explicitely in terms of geometric quantities ofF. The proof depends on a new stability result on the Aleksandrov-Fenchel inequality, which improves a theorem of Schneider.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bonnesen, T., Fenchel, W.: Theorie der konvexen Körper. Berlin: Springer. 1934.
Busemann, H.: Convex Surfaces. New York: Interscience. 1958.
Diskant, V. I.: Theorems of stability for surfaces close to a sphere. (Russian). Sibirsk. Mat. Zh.6, 1254–1266 (1965).
Diskant, V. I.: Convex surfaces with bounded mean curvature. (Russian). Sibirsk. Mat. Zh.12, 659–663 (1971). English translation: Siberian Math. J.12, 469–472 (1971).
Goodey, P. R., Groemer, H.: Stability results for first order projection bodies. Proc. Amer. Math. Soc.109, 1103–1114 (1990).
Groemer, H., Schneider, R.: Stability estimates for some geometric inequalities. Bull. London Math. Soc. To appear.
Koutroufiotis, D.: Ovaloids which are almost spheres. Comm. Pure Appl. Math.24, 289–300 (1971).
Moore, J. D.: Almost spherical convex hypersurfaces. Trans. Amer. Math. Soc.180, 347–358 (1973).
Schneider, R.: Stability in the Aleksandrov-Fenchel-Jessen Theorem. Mathematika36, 50–59 (1989).
Schneider, R.: On the Aleksandrov-Fenchel inequality for convex bodies, I. Results Math.17, 287–295 (1990).
Schneider, R.: A stability estimate for the Aleksandrov-Fenchel inequality, with an application to mean curvature. Manuscripta Math.69, 291–300 (1990).
Treibergs, A.: Existence and convexity for hyperspheres of prescribed mean curvature. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)12, 225–241 (1985).
Vitale, R. A.:L p metrics for compact convex sets. J. Approx. Theory45, 280–287 (1985).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Arnold, R. On the Aleksandrov-Fenchel inequality and the stability of the sphere. Monatshefte für Mathematik 115, 1–11 (1993). https://doi.org/10.1007/BF01311206
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01311206