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).

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
52A40

Retrieve articles in all journals with MSC (2000): 52A40

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