On the -Minkowski problem
Authors:
Erwin Lutwak, Deane Yang and Gaoyong Zhang
Translated by:
Journal:
Trans. Amer. Math. Soc. 356 (2004), 4359-4370
MSC (2000):
Primary 52A40
DOI:
https://doi.org/10.1090/S0002-9947-03-03403-2
Published electronically:
December 15, 2003
MathSciNet review:
2067123
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: A volume-normalized formulation of the -Minkowski problem is presented. This formulation has the advantage that a solution is possible for all
, including the degenerate case where the index
is equal to the dimension of the ambient space. A new approach to the
-Minkowski problem is presented, which solves the volume-normalized formulation for even data and all
.
- [A
] A. D. Aleksandrov, On the theory of mixed volumes. I. Extension of certain concepts in the theory of convex bodies, Mat. Sbornik N.S. 2 (1937), 947-972 (Russian; German summary).
- [A
] A. D. Aleksandrov, On the theory of mixed volumes. III. Extension of two theorems of Minkowski on convex polyhedra to arbitrary convex bodies, Mat. Sbornik N.S. 3 (1938), 27-46 (Russian; German summary).
- [A
] A. D. Aleksandrov, On the surface area measure of convex bodies, Mat. Sbornik N.S. 6 (1939), 167-174 (Russian; German summary). MR 1:265b
- [A
] A. D. Aleksandrov, Smoothness of the convex surface of bounded Gaussian curvature, C.R. (Dokl.) Acad. Sci. URSS 36 (1942), 195-199. MR 4:169d
- [BZ] Yu. D. Burago and V. A. Zalgaller, Geometric inequalities, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 285, Springer-Verlag, Berlin, 1988. Translated from the Russian by A. B. Sosinskiĭ; Springer Series in Soviet Mathematics. MR 936419
- [Cal] Eugenio Calabi, Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens, Michigan Math. J. 5 (1958), 105–126. MR 106487
- [CNS] L. Caffarelli, L. Nirenberg, and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. I. Monge-Ampère equation, Comm. Pure Appl. Math. 37 (1984), no. 3, 369–402. MR 739925, https://doi.org/10.1002/cpa.3160370306
- [ChY] Shiu Yuen Cheng and Shing Tung Yau, On the regularity of the solution of the 𝑛-dimensional Minkowski problem, Comm. Pure Appl. Math. 29 (1976), no. 5, 495–516. MR 423267, https://doi.org/10.1002/cpa.3160290504
- [FJ] W. Fenchel and B. Jessen, Mengenfunktionen und konvexe Körper, Danske Vid. Selskab. Mat.-fys. Medd. 16 (1938), 1-31.
- [Fi] William J. Firey, Shapes of worn stones, Mathematika 21 (1974), 1–11. MR 362045, https://doi.org/10.1112/S0025579300005714
- [G] Richard J. Gardner, Geometric tomography, Encyclopedia of Mathematics and its Applications, vol. 58, Cambridge University Press, Cambridge, 1995. MR 1356221
- [Gl
] Herman Gluck, The generalized Minkowski problem in differential geometry in the large, Ann. of Math. (2) 96 (1972), 245–276. MR 309021, https://doi.org/10.2307/1970788
- [Gl
] Herman Gluck, Manifolds with preassigned curvature—a survey, Bull. Amer. Math. Soc. 81 (1975), 313–329. MR 367861, https://doi.org/10.1090/S0002-9904-1975-13740-2
- [Le
] H. Lewy, On the existence of a closed convex surface realizing a given Riemannian metric, Proc. Nat. Acad. Sci. USA 24 (1938), 104-106.
- [Le
] H. Lewy, On differential geometry in the large, I (Minkowski's problem), Trans. Amer. Math. Soc. 43 (1938), 258-270.
- [Lu
] Erwin Lutwak, The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem, J. Differential Geom. 38 (1993), no. 1, 131–150. MR 1231704
- [Lu
] Erwin Lutwak, The Brunn-Minkowski-Firey theory. II. Affine and geominimal surface areas, Adv. Math. 118 (1996), no. 2, 244–294. MR 1378681, https://doi.org/10.1006/aima.1996.0022
- [LuO] Erwin Lutwak and Vladimir Oliker, On the regularity of solutions to a generalization of the Minkowski problem, J. Differential Geom. 41 (1995), no. 1, 227–246. MR 1316557
- [LuYZ
] E. Lutwak, D. Yang, and G. Zhang, Sharp affine
Sobolev inequalities, J. Differential Geom. 62 (2002), 17-38.
- [LuYZ
] E. Lutwak, D. Yang, and G. Zhang,
John ellipsoids, (manuscript).
- [M
] H. Minkowski, Allgemeine Lehrsätze über die konvexen Polyeder, Nachr. Ges. Wiss. Göttingen (1897), 198-219.
- [M
] H. Minkowski, Volumen und Oberfläche, Math. Ann. 57 (1903), 447-495.
- [N] L. Nirenberg, The Weyl and Minkowski problems in differential geometry in the large, Comm. Pure Appl. Math. 6 (1953), 337-394. MR 15:347b
- [P
] A.V. Pogorelov, Regularity of a convex surface with given Gaussian curvature, Mat. Sb. 31 (1952), 88-103 (Russian). MR 14:679b
- [P
] A. V. Pogorelov, A regular solution of the 𝑛-dimensional Minkowski problem, Soviet Math. Dokl. 12 (1971), 1192–1196. MR 0284956
- [P
] Aleksey Vasil′yevich Pogorelov, The Minkowski multidimensional problem, V. H. Winston & Sons, Washington, D.C.; Halsted Press [John Wiley & Sons], New York-Toronto-London, 1978. Translated from the Russian by Vladimir Oliker; Introduction by Louis Nirenberg; Scripta Series in Mathematics. MR 0478079
- [S] Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1993. MR 1216521
- [Si] David Singer, Preassigning curvature of polyhedra homeomorphic to the two-sphere, J. Differential Geometry 9 (1974), 633–638. MR 394697
- [St
] Alina Stancu, The discrete planar 𝐿₀-Minkowski problem, Adv. Math. 167 (2002), no. 1, 160–174. MR 1901250, https://doi.org/10.1006/aima.2001.2040
- [St
] A. Stancu, On the number of solutions to the discrete two-dimensional
-Minkowski problem., Adv. Math. (to appear).
- [T] A. C. Thompson, Minkowski geometry, Encyclopedia of Mathematics and its Applications, vol. 63, Cambridge University Press, Cambridge, 1996. MR 1406315
- [U]
V. Umanskiy, On solvability of the two dimensional
-Minkowski problem, Adv. Math. (to appear).
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Additional Information
Erwin Lutwak
Affiliation:
Department of Mathematics, Polytechnic University, Brooklyn, New York 11201
Email:
elutwak@poly.edu
Deane Yang
Affiliation:
Department of Mathematics, Polytechnic University, Brooklyn, New York 11201
Email:
dyang@poly.edu
Gaoyong Zhang
Affiliation:
Department of Mathematics, Polytechnic University, Brooklyn, New York 11201
Email:
gzhang@poly.edu
DOI:
https://doi.org/10.1090/S0002-9947-03-03403-2
Received by editor(s):
May 16, 2001
Received by editor(s) in revised form:
April 16, 2003
Published electronically:
December 15, 2003
Additional Notes:
This research was supported, in part, by NSF Grants DMS–9803261 and DMS–0104363
Article copyright:
© Copyright 2003
American Mathematical Society