𝐿_{𝑝}-Blaschke valuations

Li, Jin; Yuan, Shufeng; Leng, Gangsong

doi:10.1090/S0002-9947-2015-06047-4

$L_p$-Blaschke valuations
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by Jin Li, Shufeng Yuan and Gangsong Leng PDF

Trans. Amer. Math. Soc. 367 (2015), 3161-3187 Request permission

Abstract:

In this article, a classification of continuous, linearly intertwining, symmetric $L_p$-Blaschke ($p>1$) valuations is established as an extension of Haberl’s work on Blaschke valuations. More precisely, we show that for dimensions $n \geq 3$, the only continuous, linearly intertwining, normalized symmetric $L_p$-Blaschke valuation is the normalized $L_p$-curvature image operator, while for dimension $n = 2$, a rotated normalized $L_p$-curvature image operator is the only additional one. One of the advantages of our approach is that we deal with normalized symmetric $L_p$-Blaschke valuations, which makes it possible to handle the case $p=n$. The cases where $p \not =n$ are also discussed by studying the relations between symmetric $L_p$-Blaschke valuations and normalized ones.

References

S. Alesker, Continuous rotation invariant valuations on convex sets, Ann. of Math. (2) 149 (1999), no. 3, 977–1005. MR 1709308, DOI 10.2307/121078
S. Alesker, Description of translation invariant valuations on convex sets with solution of P. McMullen’s conjecture, Geom. Funct. Anal. 11 (2001), no. 2, 244–272. MR 1837364, DOI 10.1007/PL00001675
Andreas Bernig, A Hadwiger-type theorem for the special unitary group, Geom. Funct. Anal. 19 (2009), no. 2, 356–372. MR 2545241, DOI 10.1007/s00039-009-0008-4
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, 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, Blaschke valuations, Amer. J. Math. 133 (2011), no. 3, 717–751. MR 2808330, DOI 10.1353/ajm.2011.0019
Christoph Haberl, Minkowski valuations intertwining with the special linear group, J. Eur. Math. Soc. (JEMS) 14 (2012), no. 5, 1565–1597. MR 2966660, DOI 10.4171/JEMS/341
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
Christoph Haberl and Franz E. Schuster, General $L_p$ affine isoperimetric inequalities, J. Differential Geom. 83 (2009), no. 1, 1–26. MR 2545028
H. Hadwiger, Additive Funktionale $k$-dimensionaler Eikörper. I, Arch. Math. 3 (1952), 470–478 (German). MR 55707, DOI 10.1007/BF01900564
H. Hadwiger, Vorlesungen über Inhalt, Oberfläche und Isoperimetrie, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1957 (German). MR 0102775, DOI 10.1007/978-3-642-94702-5
Daniel A. Klain and Gian-Carlo Rota, Introduction to geometric probability, Lezioni Lincee. [Lincei Lectures], Cambridge University Press, Cambridge, 1997. MR 1608265
Alexander Koldobsky, Inverse formula for the Blaschke-Levy representation, Houston J. Math. 23 (1997), no. 1, 95–108. MR 1688843
Alexander L. Koldobsky, Generalized Lévy representation of norms and isometric embeddings into $L_p$-spaces, Ann. Inst. H. Poincaré Probab. Statist. 28 (1992), no. 3, 335–353 (English, with English and French summaries). MR 1183989
Kurt Leichtweiß, Affine geometry of convex bodies, Johann Ambrosius Barth Verlag, Heidelberg, 1998. MR 1630116
An Min Li, Udo Simon, and Guo Song Zhao, Global affine differential geometry of hypersurfaces, De Gruyter Expositions in Mathematics, vol. 11, Walter de Gruyter & Co., Berlin, 1993. MR 1257186, DOI 10.1515/9783110870428
Yossi Lonke, Derivatives of the $L^p$-cosine transform, Adv. Math. 176 (2003), no. 2, 175–186. MR 1982881, DOI 10.1016/S0001-8708(03)00126-9
Monika Ludwig, Projection bodies and valuations, Adv. Math. 172 (2002), no. 2, 158–168. MR 1942402, DOI 10.1016/S0001-8708(02)00021-X
Monika Ludwig, Ellipsoids and matrix-valued valuations, Duke Math. J. 119 (2003), no. 1, 159–188. MR 1991649, DOI 10.1215/S0012-7094-03-11915-8
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, Intersection bodies and valuations, Amer. J. Math. 128 (2006), no. 6, 1409–1428. MR 2275906, DOI 10.1353/ajm.2006.0046
Monika Ludwig and Matthias Reitzner, A classification of $\textrm {SL}(n)$ invariant valuations, Ann. of Math. (2) 172 (2010), no. 2, 1219–1267. MR 2680490, DOI 10.4007/annals.2010.172.1223
Erwin Lutwak, On some affine isoperimetric inequalities, J. Differential Geom. 23 (1986), no. 1, 1–13. MR 840399
Erwin Lutwak, Centroid bodies and dual mixed volumes, Proc. London Math. Soc. (3) 60 (1990), no. 2, 365–391. MR 1031458, DOI 10.1112/plms/s3-60.2.365
Erwin Lutwak, Extended affine surface area, Adv. Math. 85 (1991), no. 1, 39–68. MR 1087796, DOI 10.1016/0001-8708(91)90049-D
Erwin Lutwak, The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem, J. Differential Geom. 38 (1993), no. 1, 131–150. MR 1231704
Erwin Lutwak, Selected affine isoperimetric inequalities, Handbook of convex geometry, Vol. A, B, North-Holland, Amsterdam, 1993, pp. 151–176. MR 1242979, DOI 10.1016/B978-0-444-89596-7.50010-9
Erwin Lutwak, The Brunn-Minkowski-Firey theory. II. Affine and geominimal surface areas, Adv. Math. 118 (1996), no. 2, 244–294. MR 1378681, DOI 10.1006/aima.1996.0022
Erwin Lutwak, Deane Yang, and Gaoyong Zhang, $L_p$ affine isoperimetric inequalities, J. Differential Geom. 56 (2000), no. 1, 111–132. MR 1863023
Erwin Lutwak, Deane Yang, and Gaoyong Zhang, On the $L_p$-Minkowski problem, Trans. Amer. Math. Soc. 356 (2004), no. 11, 4359–4370. MR 2067123, DOI 10.1090/S0002-9947-03-03403-2
Peter McMullen, Valuations and dissections, Handbook of convex geometry, Vol. A, B, North-Holland, Amsterdam, 1993, pp. 933–988. MR 1243000
Peter McMullen and Rolf Schneider, Valuations on convex bodies, Convexity and its applications, Birkhäuser, Basel, 1983, pp. 170–247. MR 731112
Abraham Neyman, Representation of $L_p$-norms and isometric embedding in $L_p$-spaces, Israel J. Math. 48 (1984), no. 2-3, 129–138. MR 770695, DOI 10.1007/BF02761158
Lukas Parapatits, $\textrm {SL}(n)$-contravariant $L_p$-Minkowski valuations, Trans. Amer. Math. Soc. 366 (2014), no. 3, 1195–1211. MR 3145728, DOI 10.1090/S0002-9947-2013-05750-9
Lukas Parapatits, $\textrm {SL}(n)$-covariant $L_p$-Minkowski valuations, J. Lond. Math. Soc. (2) 89 (2014), no. 2, 397–414. MR 3188625, DOI 10.1112/jlms/jdt068
Lukas Parapatits and Franz E. Schuster, The Steiner formula for Minkowski valuations, Adv. Math. 230 (2012), no. 3, 978–994. MR 2921168, DOI 10.1016/j.aim.2012.03.024
Boris Rubin, Inversion of fractional integrals related to the spherical Radon transform, J. Funct. Anal. 157 (1998), no. 2, 470–487. MR 1638340, DOI 10.1006/jfan.1998.3268
Boris Rubin, Intersection bodies and generalized cosine transforms, Adv. Math. 218 (2008), no. 3, 696–727. MR 2414319, DOI 10.1016/j.aim.2008.01.011
Rolf Schneider, Equivariant endomorphisms of the space of convex bodies, Trans. Amer. Math. Soc. 194 (1974), 53–78. MR 353147, DOI 10.1090/S0002-9947-1974-0353147-1
Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1993. MR 1216521, DOI 10.1017/CBO9780511526282
Rolf Schneider and Franz E. Schuster, Rotation equivariant Minkowski valuations, Int. Math. Res. Not. , posted on (2006), Art. ID 72894, 20. MR 2272092, DOI 10.1155/IMRN/2006/72894
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, 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