The log-Minkowski inequality of curvature entropy. (English) Zbl 1519.52004
In calculus, the logarithm sometimes behaves as a “formal zero power”. This also happens in convexity theory, allowing some results to be generalized: in particular, we have the log-Brunn-Minkowski inequality and logarithmic Minkowski inequality of Böröczky et al.
Logarithms are also important in the definition of entropy, originally in thermodynamics (since the 19th century) and then in information theory and statistics. The various definitions have in common a formula along the lines of \(-\int f(x) \log(f(x)) dx\) and an interpretation as a measure of information content.
The authors introduce a curvature entropy for \(K,L \in\mathcal{K}^n_0\), defined as \[E(K,L) := - \int_{S^{n-1}} \log\frac{H_{n-1}(L)}{H_{n-1}(K)} dV_k ,\] where \(H\) is the Gaussian curvature of the boundary. They prove that for \(C^2\) bodies in the plane, \[E(K,L)) \leq \frac{V(K)}{2} \log\frac{V(L)}{V(K)},\] with equality only when the bodies are homothetic. They apply this to show that plane origin-symmetric \(C^2\) strictly-convex bodies with the same cone-volume measure must be equal, and relate the inequality above to other inequalities in the literature.
Logarithms are also important in the definition of entropy, originally in thermodynamics (since the 19th century) and then in information theory and statistics. The various definitions have in common a formula along the lines of \(-\int f(x) \log(f(x)) dx\) and an interpretation as a measure of information content.
The authors introduce a curvature entropy for \(K,L \in\mathcal{K}^n_0\), defined as \[E(K,L) := - \int_{S^{n-1}} \log\frac{H_{n-1}(L)}{H_{n-1}(K)} dV_k ,\] where \(H\) is the Gaussian curvature of the boundary. They prove that for \(C^2\) bodies in the plane, \[E(K,L)) \leq \frac{V(K)}{2} \log\frac{V(L)}{V(K)},\] with equality only when the bodies are homothetic. They apply this to show that plane origin-symmetric \(C^2\) strictly-convex bodies with the same cone-volume measure must be equal, and relate the inequality above to other inequalities in the literature.
Reviewer: Robert Dawson (Halifax)
MSC:
| 52A39 | Mixed volumes and related topics in convex geometry |
| 52A40 | Inequalities and extremum problems involving convexity in convex geometry |
References:
| [1] | Alexandroff, A., Existence and uniqueness of a convex surface with a given integral curvature, C. R. (Doklady) Acad. Sci. URSS (N.S.), 131-134 (1942) · Zbl 0061.37604 |
| [2] | Andrews, Ben, Classification of limiting shapes for isotropic curve flows, J. Amer. Math. Soc., 443-459 (2003) · Zbl 1023.53051 · doi:10.1090/S0894-0347-02-00415-0 |
| [3] | Andrews, Ben, Gauss curvature flow: the fate of the rolling stones, Invent. Math., 151-161 (1999) · Zbl 0936.35080 · doi:10.1007/s002220050344 |
| [4] | B\"{o}r\"{o}czky, K\'{a}roly J., On the discrete logarithmic Minkowski problem, Int. Math. Res. Not. IMRN, 1807-1838 (2016) · Zbl 1345.52002 · doi:10.1093/imrn/rnv189 |
| [5] | B\"{o}r\"{o}czky, K\'{a}roly J., Cone-volume measure of general centered convex bodies, Adv. Math., 703-721 (2016) · Zbl 1334.52003 · doi:10.1016/j.aim.2015.09.021 |
| [6] | B\"{o}r\"{o}czky, K\'{a}roly J., The logarithmic Minkowski problem, J. Amer. Math. Soc., 831-852 (2013) · Zbl 1272.52012 · doi:10.1090/S0894-0347-2012-00741-3 |
| [7] | B\"{o}r\"{o}czky, K\'{a}roly J., The log-Brunn-Minkowski inequality, Adv. Math., 1974-1997 (2012) · Zbl 1258.52005 · doi:10.1016/j.aim.2012.07.015 |
| [8] | B\"{o}r\"{o}czky, K\'{a}roly J., The dual Minkowski problem for symmetric convex bodies, Adv. Math., 106805, 30 pp. (2019) · Zbl 1427.52006 · doi:10.1016/j.aim.2019.106805 |
| [9] | B\"{o}r\"{o}czky, K\'{a}roly J., The logarithmic Minkowski problem, J. Amer. Math. Soc., 831-852 (2013) · Zbl 1272.52012 · doi:10.1090/S0894-0347-2012-00741-3 |
| [10] | Chen, Chuanqiang, Smooth solutions to the \(L_p\) dual Minkowski problem, Math. Ann., 953-976 (2019) · Zbl 1417.52008 · doi:10.1007/s00208-018-1727-3 |
| [11] | Chen, Shibing, The logarithmic Minkowski problem for non-symmetric measures, Trans. Amer. Math. Soc., 2623-2641 (2019) · Zbl 1406.52018 · doi:10.1090/tran/7499 |
| [12] | Cheng, Shiu Yuen, On the regularity of the solution of the \(n\)-dimensional Minkowski problem, Comm. Pure Appl. Math., 495-516 (1976) · Zbl 0363.53030 · doi:10.1002/cpa.3160290504 |
| [13] | Cianchi, Andrea, A unified approach to Cram\'{e}r-Rao inequalities, IEEE Trans. Inform. Theory, 643-650 (2014) · Zbl 1364.94228 · doi:10.1109/TIT.2013.2284498 |
| [14] | Colesanti, Andrea, On the stability of Brunn-Minkowski type inequalities, J. Funct. Anal., 1120-1139 (2017) · Zbl 1369.52013 · doi:10.1016/j.jfa.2017.04.008 |
| [15] | Firey, William J., Shapes of worn stones, Mathematika, 1-11 (1974) · Zbl 0311.52003 · doi:10.1112/S0025579300005714 |
| [16] | Gage, Michael E., Evolving plane curves by curvature in relative geometries, Duke Math. J., 441-466 (1993) · Zbl 0798.53041 · doi:10.1215/S0012-7094-93-07216-X |
| [17] | Gardner, Richard J., Geometric tomography, Encyclopedia of Mathematics and its Applications, xxii+492 pp. (2006), Cambridge University Press, New York · Zbl 1102.52002 · doi:10.1017/CBO9781107341029 |
| [18] | Green, Mark, Steiner polynomials, Wulff flows, and some new isoperimetric inequalities for convex plane curves, Asian J. Math., 659-676 (1999) · Zbl 0969.53040 · doi:10.4310/AJM.1999.v3.n3.a5 |
| [19] | Gruber, Peter M., Convex and discrete geometry, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], xiv+578 pp. (2007), Springer, Berlin · Zbl 1139.52001 |
| [20] | Guan, Pengfei, Entropy and a convergence theorem for Gauss curvature flow in high dimension, J. Eur. Math. Soc. (JEMS), 3735-3761 (2017) · Zbl 1386.35180 · doi:10.4171/JEMS/752 |
| [21] | Henk, Martin, Cone-volume measures of polytopes, Adv. Math., 50-62 (2014) · Zbl 1308.52007 · doi:10.1016/j.aim.2013.11.015 |
| [22] | Hu, Jiaqi, A new affine invariant geometric functional for polytopes and its associated affine isoperimetric inequalities, Int. Math. Res. Not. IMRN, 8977-8995 (2021) · Zbl 1495.52014 · doi:10.1093/imrn/rnz090 |
| [23] | Huang, Yong, On the \(L_p\) dual Minkowski problem, Adv. Math., 57-84 (2018) · Zbl 1393.52007 · doi:10.1016/j.aim.2018.05.002 |
| [24] | Huang, Yong, The Minkowski problem in Gaussian probability space, Adv. Math., Paper No. 107769, 36 pp. (2021) · Zbl 1466.52010 · doi:10.1016/j.aim.2021.107769 |
| [25] | Lutwak, Erwin, Extensions of Fisher information and Stam’s inequality, IEEE Trans. Inform. Theory, 1319-1327 (2012) · Zbl 1365.94154 · doi:10.1109/TIT.2011.2177563 |
| [26] | Lutwak, Erwin, Affine moments of a random vector, IEEE Trans. Inform. Theory, 5592-5599 (2013) · Zbl 1364.94244 · doi:10.1109/TIT.2013.2258457 |
| [27] | Lutwak, Erwin, Cram\'{e}r-Rao and moment-entropy inequalities for Renyi entropy and generalized Fisher information, IEEE Trans. Inform. Theory, 473-478 (2005) · Zbl 1205.94059 · doi:10.1109/TIT.2004.840871 |
| [28] | Lutwak, Erwin, Moment-entropy inequalities, Ann. Probab., 757-774 (2004) · Zbl 1053.60004 · doi:10.1214/aop/1079021463 |
| [29] | Ma, Lei, A new proof of the log-Brunn-Minkowski inequality, Geom. Dedicata, 75-82 (2015) · Zbl 1396.52014 · doi:10.1007/s10711-014-9979-x |
| [30] | Costa, Max H. M., On the similarity of the entropy power inequality and the Brunn-Minkowski inequality, IEEE Trans. Inform. Theory, 837-839 (1984) · Zbl 0557.94006 · doi:10.1109/TIT.1984.1056983 |
| [31] | Minkowski, Hermann, Volumen und Oberfl\"{a}che, Math. Ann., 447-495 (1903) · JFM 34.0649.01 · doi:10.1007/BF01445180 |
| [32] | H. Minkowski, Allgemeine Lehrs\"atze \"uber die konvexen, Polyeder. G\"ott. Nachr. 1897 (1897), pp. 198-219. · JFM 28.0427.01 |
| [33] | Nirenberg, Louis, The Weyl and Minkowski problems in differential geometry in the large, Comm. Pure Appl. Math., 337-394 (1953) · Zbl 0051.12402 · doi:10.1002/cpa.3160060303 |
| [34] | L. Rotem, A letter: The log-Brunn-Minkowski inequality for complex bodies, Unpublished, 1412.5321, 2014. |
| [35] | Saroglou, Christos, Remarks on the conjectured log-Brunn-Minkowski inequality, Geom. Dedicata, 353-365 (2015) · Zbl 1326.52010 · doi:10.1007/s10711-014-9993-z |
| [36] | Schneider, Rolf, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and its Applications, xiv+490 pp. (1993), Cambridge University Press, Cambridge · Zbl 0798.52001 · doi:10.1017/CBO9780511526282 |
| [37] | Stancu, Alina, The discrete planar \(L_0\)-Minkowski problem, Adv. Math., 160-174 (2002) · Zbl 1005.52002 · doi:10.1006/aima.2001.2040 |
| [38] | Stancu, Alina, The logarithmic Minkowski inequality for non-symmetric convex bodies, Adv. in Appl. Math., 43-58 (2016) · Zbl 1331.52014 · doi:10.1016/j.aam.2015.09.015 |
| [39] | Tao, Jiangyan, The logarithmic Minkowski inequality for cylinders, Proc. Amer. Math. Soc., 2143-2154 (2023) · Zbl 07660721 · doi:10.1090/proc/16307 |
| [40] | Thompson, A. C., Minkowski geometry, Encyclopedia of Mathematics and its Applications, xvi+346 pp. (1996), Cambridge University Press, Cambridge · Zbl 0868.52001 · doi:10.1017/CBO9781107325845 |
| [41] | Xi, Dongmeng, Dar’s conjecture and the log-Brunn-Minkowski inequality, J. Differential Geom., 145-189 (2016) · Zbl 1348.52006 |
| [42] | Xu, Wenxue, Entropy of chord distribution of convex bodies, Proc. Amer. Math. Soc., 3131-3141 (2019) · Zbl 1423.52006 · doi:10.1090/proc/14447 |
| [43] | Yang, Yunlong, The Green-Osher inequality in relative geometry, Bull. Aust. Math. Soc., 155-164 (2016) · Zbl 1358.52014 · doi:10.1017/S0004972715001859 |
| [44] | Zhang, Gaoyong, Proceedings of the 14th International Workshop on Differential Geometry and the 3rd KNUGRG-OCAMI Differential Geometry Workshop [Volume 14]. A lecture on integral geometry, 13-30 (2010), Natl. Inst. Math. Sci. (NIMS), Taej\u{o}n · Zbl 1217.53080 |
| [45] | Zhao, Yiming, Existence of solutions to the even dual Minkowski problem, J. Differential Geom., 543-572 (2018) · Zbl 1406.52017 · doi:10.4310/jdg/1542423629 |
| [46] | Zhao, Yiming, The dual Minkowski problem for negative indices, Calc. Var. Partial Differential Equations, Paper No. 18, 16 pp. (2017) · Zbl 1392.52005 · doi:10.1007/s00526-017-1124-x |
| [47] | Zhao, Yiming, The \(L_p\) Aleksandrov problem for origin-symmetric polytopes, Proc. Amer. Math. Soc., 4477-4492 (2019) · Zbl 1423.52012 · doi:10.1090/proc/14568 |
| [48] | Zhu, Guangxian, The logarithmic Minkowski problem for polytopes, Adv. Math., 909-931 (2014) · Zbl 1321.52015 · doi:10.1016/j.aim.2014.06.004 |
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