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Stability in the Aleksandrov-Fenchel-Jessen Theorem

Part of: General convexity

Published online by Cambridge University Press:  26 February 2010

R. Schneider
R. Schneider
Affiliation:
Mathematisches Institut, Albert-Ludwigs-Universitt, D-7800 Freiburg i. Br., FRG.
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Extract

The theorem of Aleksandrov-Fenchel-Jessen states that two convex bodies in n-dimensional Euclidean space En which, for some p l, , n - l 007D;, have equal area measures of order p (see Section 2 for a definition) differ only by a translation. Two independent proofs were given by Aleksandrov 1 and by Fenchel and Jessen 18 see also Busemann 5 (p. 70) and LeichtweiG 25 (p. 319), 26. If the boundaries of the two bodies are sufficiently smooth and of everywhere positive curvatures, then the assumption of the theorem is equivalent to saying that at points with parallel outer normals the p-th elementary symmetric functions of the principal radii of curvature of both boundary hypersurfaces are the same. For this case, Chern 6 gave a uniqueness proof by means of an integral formula.

MSC classification

Secondary: 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces)
Type
Research Article
Information
Mathematika , Volume 36 , Issue 1 , June 1989 , pp. 50 - 59
Copyright
Copyright University College London 1989

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References

1.Aleksandrov, A. D.. Zur Theorie der gemischten Volumina von konvexen Korpern. II. Neue Ungleichungen zwischen den gemischten Volumina und ihre Anwendungen. (Russian). Mat. Sbornik, N. S., 2 (1937), 12051238.Google Scholar
2.Blaschke, W.. Kreis und Kugel (Veit, Leipzig, 1916; 2nd edition: de Gruyter, Berlin, 1956).Google Scholar
3.Bonnesen, T. and Fenchel, W.. Theorie der konvexen Krper (Springer, Berlin, 1934).Google Scholar
4.Bourgain, J. and Lindenstrauss, J. Projection bodies. In: Geometric Aspects of Functional Analysis 1986-87, eds. Lindenstrauss, J. and Milman, V. D. (Lecture Notes in Math. 1317, Springer, Berlin, etc., 1988), pp. 250270.Google Scholar
5.Busemann, H.. Convex surfaces (Interscience Publ., New York, 1958).Google Scholar
6.Chern, S.-S.. Integral formulas for hypersurfaces in Euclidean space and their applications to uniqueness theorems. J. Math. Mech., 8 (1959), 947955.Google Scholar
7.Diskant, V. I.. Stability in Liebmann's theorem. (Russian). Dokl. Akad. NaukSSSR, 158 (1964), 1257-1259. English translation. Soviet Math., 5 (1964), 13871390.Google Scholar
8.Diskant, V. I.. Theorems of stability for surfaces close to a sphere. (Russian). Sibirskii Mai. , 6 (1965), 12541266.Google Scholar
9.Diskant, V. I.. Stability of a sphere in the class of convex surfaces of bounded specific curvature. (Russian). Sibirskii Mat. , 9 (1968), 816824. English translation. Siberian Math. J., 9 (1968), 610615.Google Scholar
10.Diskant, V. I.. Bounds for convex surfaces with bounded curvature functions. (Russian). Sibirskii Mat. , 12 (1971), 109125. English translation. Siberian Math. J., 12 (1971), 7889.Google Scholar
11.Diskant, V. I.. Convex surfaces with bounded mean curvature. (Russian). Sibirskii Mat. , 12 (1971), 659663. English translation. Siberian Math. J., 12 (1971), 469472Google Scholar
12.Diskant, V. I.. Bounds for the discrepancy between convex bodies in terms of the isoperimetric difference. (Russian). Sibirskii Mat. , 13 (1972), 767772.Google Scholar
13.Diskant, V. I.. Stability of a convex body under a change in the (n -2)nd curvature function. (Russian). Ukrain. Geom. Sb., 19 (1976), 2233.Google Scholar
14.Diskant, V. I.. On the question of the order of the stability function in Minkowski's problem. (Russian). Ukrain. Geom. Sb., 22 (1979), 4547.Google Scholar
15.Diskant, V. I.. Stability in Aleksandrov's problem for a convex body, one of whose projections is a ball. (Russian). Ukrain. Geom. Sb., 28 (1985), 5062.Google Scholar
16.Favard, J.. Sur les corps convexes. J. Math, pures appl. (9), 12 (1933), 21928.Google Scholar
17.Fenchel, W.. Gnralisation du thorme de Brunn et Minkowski concernant les corps convexes. C. R. Acad. Sci. Paris, 203 (1936), 764766.Google Scholar
18.Fenchel, W. and Jessen, B.. Mengenfunktionen und konvexe Korper. Danske Vid. Selsk., Math.-fys. Medd., 16, 3 (1938), 131.Google Scholar
19.Fet, A. I.. Stability theorems for convex, almost spherical surfaces. (Russian). Dokl. Akad. Nauk SSSR, 153 (1963), 537539. English translation. Soviet Math., 4 (1963), 17231725.Google Scholar
20.Fuglede, B.. Stability in the isoperimetric problem. Bull. London Math. Soc., 18 (1986), 5996.CrossRefGoogle Scholar
21.Guggenheimer, H.. Nearly spherical surfaces. Aequat. Math., 3 (1969), 186193.Google Scholar
22.Heil, E.. Extensions of an inequality of Bonnesen to D-dimensional space and curvature conditions for convex bodies. Aequat. Math., 34 (1987), 3560.Google Scholar
23.Koutroufiotis, D.. Ovaloids which are almost spheres. Commun. Pure Appl. Math., 24 (1971), 289300.Google Scholar
24.Kubota, T.. ber die Eibereiche im n-dimensionalen Raume. Science Rep. Tohoku Univ., 14 (1925), 399402.Google Scholar
25.LeichtweiB, K.. Konvexe Mengen (Springer, Berlin, etc., 1980).Google Scholar
26.LeichtweiB, K.. Zum Beweis eines Eindeutigkeitssatzes von A. D. Aleksandrov. In: B. Christoffel: The Influence of His Work on Mathematics and the Physical Sciences, eds. Butzer, P. L. and Fehr, F. (Birkhuser, Basel, 1981), pp. 636652.Google Scholar
27.Moore, J. D.. Almost spherical convex hypersurfaces. Trans. Amer. Math. Soc., 180 (1973), 347358.Google Scholar
28.Mller, C.. Spherical Harmonics (Springer, Berlin, etc., 1966).Google Scholar
29.Oliker, V. I.. On the linearized Monge-Ampre equations related to the boundary value Minkowski problem and its generalizations. In: Monge-Ampere equations and related topics (Florence, 1980) (1st. Naz. Alta Mat. Francesco Seven, Rome, 1982), pp. 79112.Google Scholar
30.Pogorelov, A. V.. Nearly spherical surfaces. J. d'Analyse math, 19 (1967), 313321.CrossRefGoogle Scholar
31.Pogorelov, A. V.. Extrinsic Geometry of Convex Surfaces (Russian; Izdat. Nauka, Moscow, 1969). English translation. (Translations of Math. Monographs, vol. 35, Amer. Math. Soc, Providence, Rhode Island, 1973).Google Scholar
32.Reetnjak, Ju. G.. Some estimates for almost umbilical surfaces. (Russian). Sibirskii Mat. , 9 (1968), 903917. English translation. Siberian Math. J., 9 (1968), 671682.Google Scholar
33.Schneider, R.. Boundary structure and curvature of convex bodies. In: Contributions to Geometry, eds. Tolke, J. and Wills, J. M. (Birkhauser, Basel, 1979), pp. 1359.CrossRefGoogle Scholar
34.Vitale, R. A.. L p metrics for compact, convex sets. J. Approx. Th., 45 (1985), 280287.Google Scholar
35.Vodop'yanov, S. K.. Estimates of the deviation from a sphere of quasi-umbilical surfaces. (Russian). Sibirskii Mat. , 11 (1970), 971987. English translation. Siberian Math. J., 11 (1970), 724735.Google Scholar
36.Volkov, Ju. A.. Stability of the solution of Minkowski's problem. (Russian). Vestnik Leningrad. Univ., Ser. Mat. Mech. Astronom., 18 (1963), 3343.Google Scholar