On the 𝐿_{𝑝}-Minkowski problem

Lutwak, Erwin; Yang, Deane; Zhang, Gaoyong

doi:10.1090/S0002-9947-03-03403-2

On the $L_{p}$-Minkowski problem
HTML articles powered by AMS MathViewer

by Erwin Lutwak, Deane Yang and Gaoyong Zhang PDF

Trans. Amer. Math. Soc. 356 (2004), 4359-4370 Request permission

Abstract:

A volume-normalized formulation of the $L_{p}$-Minkowski problem is presented. This formulation has the advantage that a solution is possible for all $p\ge 1$, including the degenerate case where the index $p$ is equal to the dimension of the ambient space. A new approach to the $L_{p}$-Minkowski problem is presented, which solves the volume-normalized formulation for even data and all $p\ge 1$.

References

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. 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).
T. Venkatarayudu, The $7$-$15$ problem, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 531. MR 0000001, DOI 10.1090/gsm/058
J. J. Corliss, Upper limits to the real roots of a real algebraic equation, Amer. Math. Monthly 46 (1939), 334–338. MR 4
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, DOI 10.1007/978-3-662-07441-1
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
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, DOI 10.1002/cpa.3160370306
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, DOI 10.1002/cpa.3160290504
W. Fenchel and B. Jessen, Mengenfunktionen und konvexe Körper, Danske Vid. Selskab. Mat.-fys. Medd. 16 (1938), 1–31.
William J. Firey, Shapes of worn stones, Mathematika 21 (1974), 1–11. MR 362045, DOI 10.1112/S0025579300005714
Richard J. Gardner, Geometric tomography, Encyclopedia of Mathematics and its Applications, vol. 58, Cambridge University Press, Cambridge, 1995. MR 1356221
Herman Gluck, The generalized Minkowski problem in differential geometry in the large, Ann. of Math. (2) 96 (1972), 245–276. MR 309021, DOI 10.2307/1970788
Herman Gluck, Manifolds with preassigned curvature—a survey, Bull. Amer. Math. Soc. 81 (1975), 313–329. MR 367861, DOI 10.1090/S0002-9904-1975-13740-2
H. Lewy, On the existence of a closed convex surface realizing a given Riemannian metric, Proc. Nat. Acad. Sci. USA 24 (1938), 104–106.
H. Lewy, On differential geometry in the large, I (Minkowski’s problem), Trans. Amer. Math. Soc. 43 (1938), 258–270.
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
Erwin Lutwak, The Brunn-Minkowski-Firey theory. II. Affine and geominimal surface areas, Adv. Math. 118 (1996), no. 2, 244–294. MR 1378681, DOI 10.1006/aima.1996.0022
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
E. Lutwak, D. Yang, and G. Zhang, Sharp affine $L_{p}$ Sobolev inequalities, J. Differential Geom. 62 (2002), 17–38.
E. Lutwak, D. Yang, and G. Zhang, $L_{p}$ John ellipsoids, (manuscript).
H. Minkowski, Allgemeine Lehrsätze über die konvexen Polyeder, Nachr. Ges. Wiss. Göttingen (1897), 198–219.
H. Minkowski, Volumen und Oberfläche, Math. Ann. 57 (1903), 447–495.
Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
A. V. Pogorelov, A regular solution of the $n$-dimensional Minkowski problem, Soviet Math. Dokl. 12 (1971), 1192–1196. MR 0284956
Aleksey Vasil′yevich Pogorelov, The Minkowski multidimensional problem, Scripta Series in Mathematics, 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. MR 0478079
Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1993. MR 1216521, DOI 10.1017/CBO9780511526282
David Singer, Preassigning curvature of polyhedra homeomorphic to the two-sphere, J. Differential Geometry 9 (1974), 633–638. MR 394697
Alina Stancu, The discrete planar $L_0$-Minkowski problem, Adv. Math. 167 (2002), no. 1, 160–174. MR 1901250, DOI 10.1006/aima.2001.2040
A. Stancu, On the number of solutions to the discrete two-dimensional $L_ 0$-Minkowski problem., Adv. Math. (to appear).
A. C. Thompson, Minkowski geometry, Encyclopedia of Mathematics and its Applications, vol. 63, Cambridge University Press, Cambridge, 1996. MR 1406315, DOI 10.1017/CBO9781107325845
V. Umanskiy, On solvability of the two dimensional $L_{p}$-Minkowski problem, Adv. Math. (to appear).