VOLUME INEQUALITIES FOR THE Lp-SINE TRANSFORM OF ISOTROPIC MEASURES

Title & Authors
VOLUME INEQUALITIES FOR THE Lp-SINE TRANSFORM OF ISOTROPIC MEASURES
Guo, LuJun; Leng, Gangsong;

Abstract
For $\small{p{\geq}1}$, sharp isoperimetric inequalities for the $\small{L_p}$-sine transform of isotropic measures are established. The corresponding reverse inequalities are obtained in an asymptotically optimal form. As applications of our main results, we present volume inequalities for convex bodies which are in $\small{L_p}$ surface isotropic position.
Keywords
isotropic measure;$\small{L_p}$-sine transform;Brascamp-Lieb inequality;reverse Brascamp-Lieb inequality;Urysohn inequality;
Language
English
Cited by
1.
Optimal Sobolev norms in the affine class, Journal of Mathematical Analysis and Applications, 2016, 436, 1, 568
References
1.
K. Ball, Shadows of convex bodies, Trans. Amer. Math. Soc. 327 (1991), no. 2, 891-901.

2.
K. Ball, Volume ratios and a reverse isoperimetric inequality, J. London Math. Soc. 44 (1991), no. 2, 351-359.

3.
F. Barthe, On a reverse form of the Brascamp-Lieb inequality, Invent. Math. 134 (1998), no. 2, 335-361.

4.
F. Barthe, A continuous version of the Brascamp-Lieb inequalities, Geometric aspects of functional analysis, 53-63, Lecture Notes in Math. 1850, Springer, Berlin, 2004.

5.
Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math. 44 (1991), no. 4, 375-417.

6.
R. J. Gardner, Geometric Tomography, Second ed., Cambridge University Press, Cambridge, 2006.

7.
R. J. Gardner and A. Giannopoulos, P-cross-section bodies, Indiana Univ. Math. J. 48 (1999), no. 2, 593-613.

8.
A. Giannopoulos and V. D. Milman, Extremal problems and isotropic positions of convex bodies, Israel J. Math. 117 (2000), 29-60.

9.
A. Giannopoulos and M. Papadimitrakis, Isotropic surface area measures, Mathematika 46 (1999), no. 1, 1-13.

10.
P. Goodey and W. Weil, The determination of convex bodies from the mean of random sections, Math. Proc. Cambridge Philos. Soc. 112 (1992), no. 2, 419-430.

11.
P. Goodey and W. Weil, Local properties of intertwining operators on the sphere, Adv. Math. 227 (2011), no. 3, 1144-1164.

12.
P. Goodey and W. Weil, A uniqueness result for mean section bodies, Adv. Math. 229 (2012), no. 1, 596-601.

13.
P. Goodey and W. Weil, Sums of sections, surface area measures, and the general Minkowski problem, J. Differential Geom. 97 (2014), no. 3, 477-514.

14.
P. Goodey, V. Yaskin, and M. Yaskina, Fourier transforms and the Funk-Hecke theorem in convex geometry, J. Lond. Math. Soc. 80 (2009), no. 2, 388-404.

15.
H. Groemer, Geometric applications of Fourier series and spherical harmonics, Encyclo-pedia of Mathematics and its Applications 61, Cambridge University Press, Cambridge, 1996.

16.
P. M. Gruber, Application of an idea of Voronoi to John type problems, Adv. Math. 218 (2008), no. 2, 309-351.

17.
C. Haberl, \$L_p\$ intersection bodies, Adv. Math. 217 (2008), ni. 6, 2599-2624.

18.
C. Haberl, Star body valued valuations, Indiana Univ. Math. J. 58 (2009), no. 5, 2253-2276.

19.
C. Haberl and F. E. Schuster, General \$L_p\$ affine isoperimetric inequalities, J. Differential Geom. 83 (2009), no. 1, 1-26.

20.
Q. Huang and B. He, Volume inequalities for complex isotropic measures, Geom. Dedicata; DOI 10.1007/s10711-014-9996-9.

21.
M. Kiderlen, Blaschke- and Minkowski-endomorphisms of convex bodies, Trans. Amer. Math. Soc. 358 (2006), no. 12, 5539-5564.

22.
D. R. Lewis, Finite dimensional subspaces of \$L_p\$, Studia Math. 63 (1978), no. 2, 207-212.

23.
E. H. Lieb, Gaussian kernels have only Gaussian maximizers, Invent. Math. 102 (1990), no. 1, 179-208.

24.
M. Ludwig, Minkowski valuations, Trans. Amer. Math. Soc. 357 (2005), no. 10, 4191- 4213.

25.
E. Lutwak, The Brunn-Minkowski-Firey theory I: Mixed volumes and the Minkowski problem, J. Differential Geom. 38 (1993), no. 1, 131-150.

26.
E. Lutwak, D. Yang, and G. Zhang, A new ellipsoid associated with convex bodies, Duke Math. J. 104 (2000), no. 3, 375-390.

27.
E. Lutwak, D. Yang, and G. Zhang, \$L_p\$ affine isoperimetric inequalities, J. Diffierential Geom. 56 (2000), no. 1, 111-132.

28.
E. Lutwak, D. Yang, and G. Zhang, Volume inequalities for subspaces of \$L_p\$, J. Differential Geom. 68 (2004), no. 1, 159-184.

29.
E. Lutwak, D. Yang, and G. Zhang, \$L_p\$ John ellipsoids, Proc. London Math. Soc. 90 (2005), no. 2, 497-520.

30.
G. Maresch and F. Schuster, The sine transform of isotropic measures, Int. Math. Res. Not. 2012 (2012), no. 4, 717-739.

31.
R. J. McCann, Existence and uniqueness of monotone measure-preserving maps, Duke Math. J. 80 (1995), no. 2, 309-323.

32.
L. Parapatits, SL(n)-contravariant \$L_p\$-Minkowski valuations, Trans. Amer. Math. Soc. 366 (2014), no. 3, 1195-1211.

33.
L. Parapatits and F. E. Schuster, The Steiner formula for Minkowski valuations, Adv. Math. 230 (2012), no. 3, 978-994.

34.
C. M. Petty, Surface area of a convex body under affine transformations, Proc. Amer. Math. Soc. 12 (1961), 824-828.

35.
D. Ryabogin and A. Zvavitch, The Fourier transform and Firey projections of convex bodies, Indiana Univ. Math. J. 53 (2004), no. 3, 667-682.

36.
R. Schneider, Uber eine Integralgleichung in der Theorie der konvexen Korper, Math. Nachr. 44 (1970), 55-75.

37.
R. Schneider, Equivariant endomorphisms of the space of convex bodies, Trans. Amer. Math. Soc. 194 (1974), 53-78.

38.
R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Cambridge University Press, 1993.

39.
R. Schneider and F. E. Schuster, Rotation invariant Minkowski classes of convex bodies, Mathematika 54 (2007), no. 1-2, 1-13.

40.
F. E. Schuster, Convolutions and multiplier transformations of convex bodies, Trans. Amer. Math. Soc. 359 (2007), no. 11, 5567-5591.

41.
F. E. Schuster, Valuations and Busemann-Petty type problems, Adv. Math. 219 (2008), no. 1, 344-368.

42.
F. E. Schuster, Crofton measures and Minkowski valuations, Duke Math. J. 154 (2010), no. 1, 1-30.

43.
F. E. Schuster and M. Weberndorfer, Volume inequalities for asymmetric Wulff shapes, J. Differential Geom. 92 (2012), no. 2, 263-283.

44.
T. Wannerer, GL(n) equivariant Minkowski valuations, Indiana Univ. Math. J. 60 (2011), no. 5, 1655-1672.

45.
M.Weberndorfer, Shadow systems of asymmetric \$L_p\$ zonotopes, Adv. Math. 240 (2013), 613-635.

46.
V. Yaskin and M. Yaskina, Centroid bodies and comparison of volumes, Indiana Univ. Math. J. 55 (2006), no. 3, 1175-1194.