On radial and polar Blaschke-Minkowski homomorphisms

Zhao, Chang-jian

doi:10.1090/S0002-9939-2012-11318-8

On radial and polar Blaschke-Minkowski homomorphisms
HTML articles powered by AMS MathViewer

by Chang-jian Zhao

Proc. Amer. Math. Soc. 141 (2013), 667-676

DOI: https://doi.org/10.1090/S0002-9939-2012-11318-8

Published electronically: May 31, 2012

PDF | Request permission

Abstract:

Brunn-Minkowski type inequalities for radial Blaschke-Minkowski homomorphisms of star bodies and polar Blaschke-Minkowski homomorphisms of convex bodies are established.

References

J. Abardia, A. Bernig, Projection bodies in complex vector spaces, Adv. Math., 227(2011), 830-846.
Semyon Alesker, Andreas Bernig, and Franz E. Schuster, Harmonic analysis of translation invariant valuations, Geom. Funct. Anal. 21 (2011), no. 4, 751–773. MR 2827009, DOI 10.1007/s00039-011-0125-8
Edwin F. Beckenbach and Richard Bellman, Inequalities, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Band 30, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961. MR 0158038, DOI 10.1007/978-3-642-64971-4
Melvin Dresher, Moment spaces and inequalities, Duke Math. J. 20 (1953), 261–271. MR 55389
Christoph Haberl, Star body valued valuations, Indiana Univ. Math. J. 58 (2009), no. 5, 2253–2276. MR 2583498, DOI 10.1512/iumj.2009.58.3685
Christoph Haberl and Franz E. Schuster, General $L_p$ affine isoperimetric inequalities, J. Differential Geom. 83 (2009), no. 1, 1–26. MR 2545028
Christoph Haberl, $L_p$ intersection bodies, Adv. Math. 217 (2008), no. 6, 2599–2624. MR 2397461, DOI 10.1016/j.aim.2007.11.013
Christoph Haberl and Monika Ludwig, A characterization of $L_p$ intersection bodies, Int. Math. Res. Not. , posted on (2006), Art. ID 10548, 29. MR 2250020, DOI 10.1155/IMRN/2006/10548
Markus Kiderlen, Blaschke- and Minkowski-endomorphisms of convex bodies, Trans. Amer. Math. Soc. 358 (2006), no. 12, 5539–5564. MR 2238926, DOI 10.1090/S0002-9947-06-03914-6
Monika Ludwig, Minkowski valuations, Trans. Amer. Math. Soc. 357 (2005), no. 10, 4191–4213. MR 2159706, DOI 10.1090/S0002-9947-04-03666-9
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, Minkowski areas and valuations, J. Differential Geom. 86 (2010), no. 1, 133–161. MR 2772547
Erwin Lutwak, Intersection bodies and dual mixed volumes, Adv. in Math. 71 (1988), no. 2, 232–261. MR 963487, DOI 10.1016/0001-8708(88)90077-1
Erwin Lutwak, Dual mixed volumes, Pacific J. Math. 58 (1975), no. 2, 531–538. MR 380631, DOI 10.2140/pjm.1975.58.531
Erwin Lutwak, Mixed projection inequalities, Trans. Amer. Math. Soc. 287 (1985), no. 1, 91–105. MR 766208, DOI 10.1090/S0002-9947-1985-0766208-7
Franz E. Schuster, Volume inequalities and additive maps of convex bodies, Mathematika 53 (2006), no. 2, 211–234 (2007). MR 2343256, DOI 10.1112/S0025579300000103
Franz E. Schuster, Valuations and Busemann-Petty type problems, Adv. Math. 219 (2008), no. 1, 344–368. MR 2435426, DOI 10.1016/j.aim.2008.05.001
Franz E. Schuster, Convolutions and multiplier transformations of convex bodies, Trans. Amer. Math. Soc. 359 (2007), no. 11, 5567–5591. MR 2327043, DOI 10.1090/S0002-9947-07-04270-5
Franz E. Schuster, Crofton measures and Minkowski valuations, Duke Math. J. 154 (2010), no. 1, 1–30. MR 2668553, DOI 10.1215/00127094-2010-033
Franz E. Schuster and Thomas Wannerer, $\textrm {GL}(n)$ contravariant Minkowski valuations, Trans. Amer. Math. Soc. 364 (2012), no. 2, 815–826. MR 2846354, DOI 10.1090/S0002-9947-2011-05364-X
T. Wannerer, $GL(n)$ equivariant Minkowski valuations, Indiana Univ. Math. J., in press. http://www.iumj.indiana.edu/IUMJ/forthcoming.php
Chang-jian Zhao, On Blaschke-Minkowski homomorphisms, Geom. Dedicata 149 (2010), 373–378. MR 2737698, DOI 10.1007/s10711-010-9487-6
Chang-jian Zhao, On intersection and mixed intersection bodies, Geom. Dedicata 141 (2009), 109–122. MR 2520066, DOI 10.1007/s10711-008-9346-x
ChangJian Zhao, $L_p$-mixed intersection bodies, Sci. China Ser. A 51 (2008), no. 12, 2172–2188. MR 2462021, DOI 10.1007/s11425-008-0074-3
Chang-jian Zhao, $L_p$-dual quermassintegral sums, Sci. China Ser. A 50 (2007), no. 9, 1347–1360. MR 2370622, DOI 10.1007/s11425-007-0117-1
Chang-jian Zhao and Gang-song Leng, On polars of mixed projection bodies, J. Math. Anal. Appl. 316 (2006), no. 2, 664–678. MR 2207338, DOI 10.1016/j.jmaa.2005.04.073
Chang-jian Zhao and Gangsong Leng, Brunn-Minkowski inequality for mixed intersection bodies, J. Math. Anal. Appl. 301 (2005), no. 1, 115–123. MR 2105924, DOI 10.1016/j.jmaa.2004.07.013