The 𝐿_{𝑝} Aleksandrov problem for origin-symmetric polytopes

Zhao, Yiming

doi:10.1090/proc/14568

The $L_p$ Aleksandrov problem for origin-symmetric polytopes
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by Yiming Zhao PDF

Proc. Amer. Math. Soc. 147 (2019), 4477-4492 Request permission

Abstract:

The $L_p$ Aleksandrov integral curvature and its corresponding characterization problem, the $L_p$ Aleksandrov problem, were recently introduced by Huang, Lutwak, Yang, and Zhang. The current work presents a solution to the $L_p$ Aleksandrov problem for origin-symmetric polytopes when $-1<p<0$.

References

A. Alexandroff, Existence and uniqueness of a convex surface with a given integral curvature, C. R. (Doklady) Acad. Sci. URSS (N.S.) 35 (1942), 131–134. MR 0007625
Ben Andrews, Gauss curvature flow: the fate of the rolling stones, Invent. Math. 138 (1999), no. 1, 151–161. MR 1714339, DOI 10.1007/s002220050344
Ben Andrews, Classification of limiting shapes for isotropic curve flows, J. Amer. Math. Soc. 16 (2003), no. 2, 443–459. MR 1949167, DOI 10.1090/S0894-0347-02-00415-0
Jérôme Bertrand, Prescription of Gauss curvature using optimal mass transport, Geom. Dedicata 183 (2016), 81–99. MR 3523119, DOI 10.1007/s10711-016-0147-3
K. J. Böröczky and F. Fodor, The ${L}_p$ dual Minkowski problem for $p>1$ and $q>0$, J. Differential Equations 266 (2019), 7980–8033, DOI 10.1016/j.jde.2018.12.020.
Károly J. Böröczky, Pál Hegedűs, and Guangxian Zhu, On the discrete logarithmic Minkowski problem, Int. Math. Res. Not. IMRN 6 (2016), 1807–1838. MR 3509941, DOI 10.1093/imrn/rnv189
Károly J. Böröczky and Martin Henk, Cone-volume measure of general centered convex bodies, Adv. Math. 286 (2016), 703–721. MR 3415694, DOI 10.1016/j.aim.2015.09.021
Károly J. Böröczky, Martin Henk, and Hannes Pollehn, Subspace concentration of dual curvature measures of symmetric convex bodies, J. Differential Geom. 109 (2018), no. 3, 411–429. MR 3825606, DOI 10.4310/jdg/1531188189
Károly J. Böröczky, Erwin Lutwak, Deane Yang, and Gaoyong Zhang, The log-Brunn-Minkowski inequality, Adv. Math. 231 (2012), no. 3-4, 1974–1997. MR 2964630, DOI 10.1016/j.aim.2012.07.015
Károly J. Böröczky, Erwin Lutwak, Deane Yang, and Gaoyong Zhang, The logarithmic Minkowski problem, J. Amer. Math. Soc. 26 (2013), no. 3, 831–852. MR 3037788, DOI 10.1090/S0894-0347-2012-00741-3
Károly J. Böröczky, Erwin Lutwak, Deane Yang, and Gaoyong Zhang, Affine images of isotropic measures, J. Differential Geom. 99 (2015), no. 3, 407–442. MR 3316972
K. J. Böröczky, E. Lutwak, D. Yang, G. Zhang, and Y. Zhao, The dual Minkowski problem for symmetric convex bodies, preprint.
C. Chen, Y. Huang, and Y. Zhao, Smooth solutions to the ${L}_p$ dual Minkowski problem, Math. Ann. (2018), https://doi.org/10.1007/s00208-018-1727-3.
Shibing Chen, Qi-rui Li, and Guangxian Zhu, On the $L_p$ Monge-Ampère equation, J. Differential Equations 263 (2017), no. 8, 4997–5011. MR 3680945, DOI 10.1016/j.jde.2017.06.007
Shibing Chen, Qi-rui Li, and Guangxian Zhu, The logarithmic Minkowski problem for non-symmetric measures, Trans. Amer. Math. Soc. 371 (2019), no. 4, 2623–2641. MR 3896091, DOI 10.1090/tran/7499
Wenxiong Chen, $L_p$ Minkowski problem with not necessarily positive data, Adv. Math. 201 (2006), no. 1, 77–89. MR 2204749, DOI 10.1016/j.aim.2004.11.007
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
Kai-Seng Chou and Xu-Jia Wang, The $L_p$-Minkowski problem and the Minkowski problem in centroaffine geometry, Adv. Math. 205 (2006), no. 1, 33–83. MR 2254308, DOI 10.1016/j.aim.2005.07.004
Richard J. Gardner, Daniel Hug, Wolfgang Weil, Sudan Xing, and Deping Ye, General volumes in the Orlicz-Brunn-Minkowski theory and a related Minkowski problem I, Calc. Var. Partial Differential Equations 58 (2019), no. 1, Paper No. 12, 35. MR 3882970, DOI 10.1007/s00526-018-1449-0
Christoph Haberl and Franz E. Schuster, Asymmetric affine $L_p$ Sobolev inequalities, J. Funct. Anal. 257 (2009), no. 3, 641–658. MR 2530600, DOI 10.1016/j.jfa.2009.04.009
Martin Henk and Eva Linke, Cone-volume measures of polytopes, Adv. Math. 253 (2014), 50–62. MR 3148545, DOI 10.1016/j.aim.2013.11.015
Martin Henk and Hannes Pollehn, Necessary subspace concentration conditions for the even dual Minkowski problem, Adv. Math. 323 (2018), 114–141. MR 3725875, DOI 10.1016/j.aim.2017.10.037
Yong Huang, Jiakun Liu, and Lu Xu, On the uniqueness of $L_p$-Minkowski problems: the constant $p$-curvature case in $\Bbb {R}^3$, Adv. Math. 281 (2015), 906–927. MR 3366857, DOI 10.1016/j.aim.2015.02.021
Yong Huang, Erwin Lutwak, Deane Yang, and Gaoyong Zhang, Geometric measures in the dual Brunn-Minkowski theory and their associated Minkowski problems, Acta Math. 216 (2016), no. 2, 325–388. MR 3573332, DOI 10.1007/s11511-016-0140-6
Yong Huang, Erwin Lutwak, Deane Yang, and Gaoyong Zhang, The $L_p$-Aleksandrov problem for $L_p$-integral curvature, J. Differential Geom. 110 (2018), no. 1, 1–29. MR 3851743, DOI 10.4310/jdg/1536285625
Yong Huang and Yiming Zhao, On the $L_p$ dual Minkowski problem, Adv. Math. 332 (2018), 57–84. MR 3810248, DOI 10.1016/j.aim.2018.05.002
Daniel Hug, Erwin Lutwak, Deane Yang, and Gaoyong Zhang, On the $L_p$ Minkowski problem for polytopes, Discrete Comput. Geom. 33 (2005), no. 4, 699–715. MR 2132298, DOI 10.1007/s00454-004-1149-8
Huaiyu Jian, Jian Lu, and Xu-Jia Wang, Nonuniqueness of solutions to the $L_p$-Minkowski problem, Adv. Math. 281 (2015), 845–856. MR 3366854, DOI 10.1016/j.aim.2015.05.010
Huaiyu Jian, Jian Lu, and Guangxian Zhu, Mirror symmetric solutions to the centro-affine Minkowski problem, Calc. Var. Partial Differential Equations 55 (2016), no. 2, Art. 41, 22. MR 3479715, DOI 10.1007/s00526-016-0976-9
Q. R. Li, W. M. Sheng, and X. J. Wang, Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems, J. Eur. Math. Soc. (in press).
Jian Lu and Xu-Jia Wang, Rotationally symmetric solutions to the $L_p$-Minkowski problem, J. Differential Equations 254 (2013), no. 3, 983–1005. MR 2997361, DOI 10.1016/j.jde.2012.10.008
Monika Ludwig, General affine surface areas, Adv. Math. 224 (2010), no. 6, 2346–2360. MR 2652209, DOI 10.1016/j.aim.2010.02.004
Monika Ludwig and Matthias Reitzner, A classification of $\textrm {SL}(n)$ invariant valuations, Ann. of Math. (2) 172 (2010), no. 2, 1219–1267. MR 2680490, DOI 10.4007/annals.2010.172.1223
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
Erwin Lutwak, Deane Yang, and Gaoyong Zhang, $L_p$ affine isoperimetric inequalities, J. Differential Geom. 56 (2000), no. 1, 111–132. MR 1863023
Erwin Lutwak, Deane Yang, and Gaoyong Zhang, Sharp affine $L_p$ Sobolev inequalities, J. Differential Geom. 62 (2002), no. 1, 17–38. MR 1987375
Erwin Lutwak, Deane Yang, and Gaoyong Zhang, On the $L_p$-Minkowski problem, Trans. Amer. Math. Soc. 356 (2004), no. 11, 4359–4370. MR 2067123, DOI 10.1090/S0002-9947-03-03403-2
Erwin Lutwak, Deane Yang, and Gaoyong Zhang, $L_p$ dual curvature measures, Adv. Math. 329 (2018), 85–132. MR 3783409, DOI 10.1016/j.aim.2018.02.011
Assaf Naor and Dan Romik, Projecting the surface measure of the sphere of $\scr l_p^n$, Ann. Inst. H. Poincaré Probab. Statist. 39 (2003), no. 2, 241–261 (English, with English and French summaries). MR 1962135, DOI 10.1016/S0246-0203(02)00008-0
Vladimir Oliker, Embedding $\mathbf S^n$ into $\mathbf R^{n+1}$ with given integral Gauss curvature and optimal mass transport on $\mathbf S^n$, Adv. Math. 213 (2007), no. 2, 600–620. MR 2332603, DOI 10.1016/j.aim.2007.01.005
Grigoris Paouris and Elisabeth M. Werner, Relative entropy of cone measures and $L_p$ centroid bodies, Proc. Lond. Math. Soc. (3) 104 (2012), no. 2, 253–286. MR 2880241, DOI 10.1112/plms/pdr030
Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Second expanded edition, Encyclopedia of Mathematics and its Applications, vol. 151, Cambridge University Press, Cambridge, 2014. MR 3155183
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
Ge Xiong, Extremum problems for the cone volume functional of convex polytopes, Adv. Math. 225 (2010), no. 6, 3214–3228. MR 2729006, DOI 10.1016/j.aim.2010.05.016
Yiming Zhao, The dual Minkowski problem for negative indices, Calc. Var. Partial Differential Equations 56 (2017), no. 2, Paper No. 18, 16. MR 3605843, DOI 10.1007/s00526-017-1124-x
Yiming Zhao, Existence of solutions to the even dual Minkowski problem, J. Differential Geom. 110 (2018), no. 3, 543–572. MR 3880233, DOI 10.4310/jdg/1542423629
Guangxian Zhu, The logarithmic Minkowski problem for polytopes, Adv. Math. 262 (2014), 909–931. MR 3228445, DOI 10.1016/j.aim.2014.06.004
Guangxian Zhu, The centro-affine Minkowski problem for polytopes, J. Differential Geom. 101 (2015), no. 1, 159–174. MR 3356071
Guangxian Zhu, The $L_p$ Minkowski problem for polytopes for $p<0$, Indiana Univ. Math. J. 66 (2017), no. 4, 1333–1350. MR 3689334, DOI 10.1512/iumj.2017.66.6110