In this paper, we study some basic properties on Lipschitz star bodies, such as the equivalence between Lipschitz star bodies and star bodies with respect to a ball, the equivalence between the convergence of Lipschitz star bodies with respect to Hausdorff distance and the convergence of Lipschtz star bodies with respect to radial distance, and the convergence of Steiner symmetrizations of Lipschitz star bodies.
Youjiang Lin
,
Yuchi
,
Wu
. LIPSCHITZ STAR BODIES*[J]. Acta mathematica scientia, Series B, 2023
, 43(2)
: 597
-607
.
DOI: 10.1007/s10473-023-0208-2
[1] Beer G A.The Hausdorff metric and convergence in measure. Michigan Math J, 1974, 21: 63-64
[2] Beer G A.Starshaped sets and the Hausdorff metric. Pacific J Math, 1975, 61(1): 21-27
[3] Bianchi G, Klain D A, Lutwak E, et al.A countable set of directions is sufficient for Steiner symmetrization. Adv Appl Math, 2011, 47(4): 869-873
[4] Campi S, Gronchi P.The Lp-Busemann-Petty centroid inequality. Adv Math, 2002, 167: 128-141
[5] Cormier R J.Steiner symmetrization in En. Rev Mat Hisp-Amer, 1971, 31(4): 197-204
[6] Gardner R J.On the Busemann-Petty problem concerning central sections of centrally symmetric convex bodies. Bull Amer Math Soc, 1994, 30: 222-226
[7] Gardner R J.Intersection bodies and the Busemann-Petty problem. Trans Amer Math Soc, 1994, 342: 435-445
[8] Gardner R J.A positive answer to the Busemann-Petty problem in three dimensions. Ann of Math, 1994, 140: 435-447
[9] Gardner R J.The Brunn-Minkowski inequality. Bull Amer Math Soc, 2002, 39(3): 355-405
[10] Gardner R J. Geometric Tomography.Cambridge: Cambridge University Press, 2006
[11] Gardner R J.The dual Brunn-Minkowski theory for bounded Borel sets: dual affine quermassintegrals and inequalities. Adv Math, 2007, 216: 358-386
[12] Gardner R J, Volčič A.Tomography of convex and star bodies. Adv Math, 1994, 108: 367-399
[13] Gruber P M.Convex and Discrete Geometry. Berlin: Springer, 2007
[14] Haberl C.Star body valued valuations. Indiana Univ Math J, 2009, 58(5): 2253-2276
[15] Huang Y, Lutwak E, Yang D, et al.Geometric measures in the dual Brunn-Minkowski theory and their associated Minkowski problems. Acta Math, 2016, 216: 325-388
[16] Lin Y.Smoothness of the Steiner symmetrization. Proc Amer Math Soc, 2018, 146(1): 345-357
[17] Lin Y.Affine Orlicz Pólya-Szegö principle for log-concave functions. J Funct Anal, 2017, 273: 3295-3326
[18] Ludwig M.Intersection bodies and valuations. Amer J Math, 2006, 128: 1409-1428
[19] Ludwig M, Xiao J, Zhang G.Sharp convex Lorentz-Sobolev inequalities. Math Ann, 2011, 350(1): 169-197
[20] Lutwak E.Dual mixed volumes. Pacific J Math, 1975, 58: 531-538
[21] Lutwak E.Dual cross-sectional measures. Atti Accad Naz Lincei, 1975, 58: 1-5
[22] Lutwak E, Yang D, Zhang G.Lp affine isoperimetric inequalities. J Differential Geom, 2000, 56(1): 111-132
[23] Lutwak E, Yang D, Zhang G.Orlicz centroid bodies. J Differential Geom, 2010, 84(2): 365-387
[24] Maz'ya V. Sobolev Spaces with Applications to Elliptic Partial Differential Equations. Heidelberg: Springer, 2011
[25] Schneider R.Convex Bodies, the Brunn-Minkowski Theory. Cambridge: Cambridge University Press, 2014
[26] Volčič A.On iterations of Steiner symmetrizations. Ann Mat Pura Appl, 2016, 195(5): 1685-1692
[27] Webster R.Convexity. New York: Oxford University Press, 1994
[28] Zhang G.A positive solution to the Busemann-Petty problem in $\mathbb{R}^{4}$. Ann Math, 1999, 149: 535-543
[29] Zhu G.The Orlicz centroid inequality for star bodies. Adv in Appl Math, 2012, 48(2): 432-445
