Czechoslovak Mathematical Journal, first online, pp. 1-13


Inequalities for general width-integrals of Blaschke-Minkowski homomorphisms

Chao Li, Weidong Wang

Received November 27, 2018.   Published online January 27, 2020.

Abstract:  We establish some inequalities for general width-integrals of Blaschke-Minkowski homomorphisms. As applications, inequalities for width-integrals of projection bodies are derived.
Keywords:  general width-integral; volume difference type inequality; Blaschke-Minkowski homomorphism; Brunn-Minkowski type inequality; projection body
Classification MSC:  52A20, 52A40
DOI:  10.21136/CMJ.2020.0521-18

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References:
[1] E. Beckenbach, R. Bellman: Inequalities. Ergebnisse der Mathematik und Ihrer Grenzgebiete 30, Springer, New York (1965). DOI 10.1007/978-3-642-64971-4 | MR 0192009 | Zbl 0126.28002
[2] A. Berg, L. Parapatits, F. E. Schuster, M. Weberndorfer: Log-concavity properties of Minkowski valuations. Trans. Am. Math. Soc. 370 (2018), 5245-5277. DOI 10.1090/tran/7434 | MR 3787383 | Zbl 1390.52024
[3] W. Blaschke: Vorlesungen über Integralgeometrie. VEB Deutscher Verlag der Wissenschaften, Berlin (1955). (In German.) MR 0076373 | Zbl 0066.40703
[4] W.-S. Cheung, C.-J. Zhao: Width-integrals and affine surface area of convex bodies. Banach J. Math. Anal. 2 (2008), 70-77. DOI 10.15352/bjma/1240336275 | MR 2404711 | Zbl 1155.52005
[5] M. Dresher: Moment spaces and inequalities. Duke Math. J. 20 (1953), 261-271. DOI 10.1215/s0012-7094-53-02026-2 | MR 0055389 | Zbl 0050.28202
[6] Y. Feng: General mixed width-integral of convex bodies. J. Nonlinear Sci. Appl. 9 (2016), 4226-4234. DOI 10.22436/jnsa.009.06.64 | MR 3530126 | Zbl 1347.52004
[7] Y. Feng, W. Wang: Blaschke-Minkowski homomorphisms and affine surface area. Publ. Math. 85 (2014), 297-308. DOI 10.5486/PMD.2014.5903 | MR 3291832 | Zbl 1340.52006
[8] Y. Feng, W. Wang, J. Yuan: Inequalities of quermassintegrals about mixed Blaschke Minkowski homomorphisms. Tamkang J. Math. 46 (2015), 217-227. DOI 10.5556/j.tkjm.46.2015.1689 | MR 3406352 | Zbl 1338.52005
[9] Y. Feng, S. Wu: Brunn-Minkowski type inequalies for width-integrals of index $i$. J. Comput. Anal. Appl. 24 (2018), 1408-1418. MR 3753402
[10] Y. Feng, S. Wu, W. Wang: Mixed chord-integrals of index $i$ and radial Blaschke-Minkowski homomorphisms. Rocky Mt. J. Math. 47 (2017), 2627-2640. DOI 10.1216/RMJ-2017-47-8-2627 | MR 3760310 | Zbl 1385.52005
[11] W. J. Firey: Mean cross-section measures of harmonic means of convex bodies. Pac. J. Math. 11 (1961), 1263-1266. DOI 10.2140/pjm.1961.11.1263 | MR 0140003 | Zbl 0122.41101
[12] R. J. Gardner: Geometric Tomography. Encyclopedia of Mathematics and Its Applications 58, Cambridge University Press, Cambridge (2006). DOI 10.1017/CBO9781107341029 | MR 2251886 | Zbl 1102.52002
[13] C. Haberl: Minkowski valuations intertwining with the special linear group. J. Eur. Math. Soc. (JEMS) 14 (2012), 1565-1597. DOI 10.4171/JEMS/341 | MR 2966660 | Zbl 1270.52018
[14] M. A. Hernández Cifre, J. Yepes Nicolás: On Brunn-Minkowski-type inequalities for polar bodies. J. Geom. Anal. 26 (2016), 143-155. DOI 10.1007/s12220-014-9541-y | MR 3441506 | Zbl 1339.52007
[15] L. Ji, Z. Zeng: Some inequalities for radial Blaschke-Minkowski homomorphisms. Czech. Math. J. 67 (2017), 779-793. DOI 10.21136/CMJ.2017.0180-16 | MR 3697916 | Zbl 06770130
[16] Y. Li, W. Wang: Monotonicity inequalities for $L_p$ Blaschke-Minkowski homomorphism. J. Inequal. Appl. 2014 (2014), Article ID 131, 10 pages. DOI 10.1186/1029-242X-2014-131 | MR 3253878 | Zbl 1310.52003
[17] X.-Y. Li, C.-J. Zhao: On the $p$-mixed affine surface area. Math. Inequal. Appl. 17 (2014), 443-450. DOI 10.7153/mia-17-33 | MR 3235022 | Zbl 1296.52004
[18] F. Lu, G. Leng: On inequalities for $i$th width-integrals of convex bodies. Math. Appl. 19 (2006), 632-636. (In Chinese.) MR 2254976 | Zbl 1101.52001
[19] M. Ludwig: Minkowski valuations. Trans. Am. Math. Soc. 357 (2005), 4191-4213. DOI 10.1090/S0002-9947-04-03666-9 | MR 2159706 | Zbl 1077.52005
[20] E. Lutwak: Width-integrals of convex bodies. Proc. Am. Math. Soc. 53 (1975), 435-439. DOI 10.1090/S0002-9939-1975-0383254-5 | MR 0383254 | Zbl 0276.52006
[21] E. Lutwak: Mixed width-integrals of convex bodies. Isr. J. Math. 28 (1977), 249-253. DOI 10.1007/BF02759811 | MR 0464070 | Zbl 0363.52009
[22] E. Lutwak, D. Yang, G. Zhang: Orlicz projection bodies. Adv. Math. 223 (2010), 220-242. DOI 10.1016/j.aim.2009.08.002 | MR 2563216 | Zbl 05643962
[23] S. Lv: Dual Brunn-Minkowski inequality for volume differences. Geom. Dedicata 145 (2010), 169-180. DOI 10.1007/s10711-009-9414-x | MR 2600953 | Zbl 1202.52008
[24] R. Schneider: Convex Bodies: The Brunn-Minkowski Theory. Encyclopedia of Mathematics and its Applications 151, Cambridge University Press, Cambridge (2014). DOI 10.1017/CBO9781139003858 | MR 3155183 | Zbl 1287.52001
[25] F. E. Schuster: Volume inequalities and additive maps of convex bodies. Mathematica 53 (2006), 211-234. DOI 10.1112/S0025579300000103 | MR 2343256 | Zbl 1129.52002
[26] F. E. Schuster: Valuations and Busemann-Petty type problems. Adv. Math. 219 (2008), 344-368. DOI 10.1016/j.aim.2008.05.001 | MR 2435426 | Zbl 1146.52003
[27] F. E. Schuster: Crofton measures and Minkowski valuations. Duke Math. J. 154 (2010), 1-30. DOI 10.1215/00127094-2010-033 | MR 2668553 | Zbl 1205.52004
[28] F. E. Schuster, T. Wannerer: Even Minkowski valuations. Am. J. Math. 137 (2015), 1651-1683. DOI 10.1353/ajm.2015.0041 | MR 3432270 | Zbl 1336.52020
[29] F. E. Schuster, T. Wannerer: Minkowski valuations and generalized valuations. J. Eur. Math. Soc. (JEMS) 20 (2018), 1851-1884. DOI 10.4171/JEMS/801 | MR 3854893 | Zbl 1398.52018
[30] T. Zhang, W. Wang: Inequalities for mixed width-integrals. Wuhan Univ. J. Nat. Sci. 21 (2016), 185-190. DOI 10.1007/s11859-016-1157-6 | MR 3525752 | Zbl 1363.26055
[31] C.-J. Zhao: On Blaschke-Minkowski homomorphisms. Geom. Dedicata 149 (2010), 373-378. DOI 10.1007/s10711-010-9487-6 | MR 2737698 | Zbl 1207.52009
[32] C.-J. Zhao: On polars of Blaschke-Minkowski homomorphisms. Math. Scand. 111 (2012), 147-160. DOI 10.7146/math.scand.a-15220 | MR 3001365 | Zbl 1281.52006
[33] C.-J. Zhao: Volume sums of polar Blaschke-Minkowski homomorphisms. Proc. Indian Acad. Sci., Math. Sci. 125 (2015), 209-219. DOI 10.1007/s12044-015-0227-6 | MR 3361514 | Zbl 1321.52014
[34] C.-J. Zhao: On Blaschke-Minkowski homomorphisms and radial Blaschke-Minkowski homomorphisms. J. Geom. Anal. 26 (2016), 1523-1538. DOI 10.1007/s12220-015-9598-2 | MR 3472843 | Zbl 1350.52004
[35] C.-J. Zhao, W.-S. Cheung: Radial Blaschke-Minkowski homomorphisms and volume differences. Geom. Dedicata 154 (2011), 81-91. DOI 10.1007/s10711-010-9568-6 | MR 2832712 | Zbl 1230.52023
[36] C.-J. Zhao, B. Mihály: Width-integrals of mixed projection bodies and mixed affine surface area. Gen. Math. 19 (2011), 123-133. MR 2788350 | Zbl 1224.52016
[37] Y. Zhou: General $L_p$-mixed width-integral of convex bodies and related inequalities. J. Nonlinear Sci. Appl. 10 (2017), 4372-4380. DOI 10.22436/jnsa.010.08.30 | MR 3702585 | Zbl 1412.52005

Affiliations:   Chao Li, Weidong Wang (corresponding author), Department of Mathematics, China Three Gorges University, Three Gorges Mathematical Research Center, 8 Daxue Rd, Xiling, Yichang, Hubei, P. R. China, 443002, e-mail: lichao166298@163.com, wangwd722@163.com


 
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