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Valuations on convex functions and convex sets and Monge–Ampère operators

  • Semyon Alesker EMAIL logo
Published/Copyright: July 12, 2019
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Advances in Geometry
From the journal Advances in Geometry Volume 19 Issue 3

Abstract

The notion of a valuation on convex bodies is very classical; valuations on a class of functions have been introduced and studied by M. Ludwig and others. We study an explicit relation between continuous valuations on convex functions which are invariant under adding arbitrary linear functionals, and translation invariant continuous valuations on convex bodies. More precisely, we construct a natural linear map from the former space to the latter and prove that it has dense image and infinite-dimensional kernel. The proof uses the author’s irreducibility theorem and properties of the real Monge–Ampère operators due to A.D. Alexandrov and Z. Blocki. Furthermore we show how to use complex, quaternionic, and octonionic Monge–Ampère operators to construct more examples of continuous valuations on convex functions in an analogous way.

Keywords: Valuation on convex bodies; convex set; Monge–Ampère operator
MSC 2010: 52A43

  1. Communicated by: M. Henk

  2. Funding: This work was partially supported by ISF grant 865/16.

References

[1] A. D. Aleksandrov, Dirichlet’s problem for the equation Det ∥zij∥ = φ (z1, …, zn, z, x1, …, xn). I. (Russian) Vestnik Leningrad. Univ. Ser. Mat. Meh. Astr. 13 (1958), 5–24. MR0096903 Zbl 0114.30202Search in Google Scholar

[2] S. Alesker, Description of translation invariant valuations on convex sets with solution of P. McMullen’s conjecture. Geom. Funct. Anal. 11 (2001), 244–272. MR1837364 Zbl 0995.5200110.1007/PL00001675Search in Google Scholar

[3] S. Alesker, Non-commutative linear algebra and plurisubharmonic functions of quaternionic variables. Bull. Sci. Math. 127 (2003), 1–35. MR1957796 Zbl 1033.1501310.1016/S0007-4497(02)00004-0Search in Google Scholar

[4] S. Alesker, Valuations on convex sets, non-commutative determinants, and pluripotential theory. Adv. Math. 195 (2005), 561–595. MR2146354 Zbl 1078.5201110.1016/j.aim.2004.08.009Search in Google Scholar

[5] S. Alesker, Plurisubharmonic functions on the octonionic plane and Spin(9)-invariant valuations on convex sets. J. Geom. Anal. 18 (2008), 651–686. MR2420758 Zbl 1165.3201510.1007/s12220-008-9032-0Search in Google Scholar

[6] H. Aslaksen, Quaternionic determinants. Math. Intelligencer18 (1996), 57–65. MR1412993 Zbl 0881.1500710.1007/978-1-4613-0195-0_13Search in Google Scholar

[7] A. Bernig, J. H. G. Fu, Hermitian integral geometry. Ann. of Math. (2)173 (2011), 907–945. MR2776365 Zbl 1230.5201410.4007/annals.2011.173.2.7Search in Google Scholar

[8] Z. Blocki, Equilibrium measure of a product subset of ℂn. Proc. Amer. Math. Soc. 128 (2000), 3595–3599. MR1707508 Zbl 0959.3203910.1090/S0002-9939-00-05552-0Search in Google Scholar

[9] S. S. Chern, H. I. Levine, L. Nirenberg, Intrinsic norms on a complex manifold. In: Global Analysis (Papers in Honor of K. Kodaira), 119–139, Univ. Tokyo Press, Tokyo 1969. MR0254877 Zbl 0202.11603Search in Google Scholar

[10] A. Colesanti, M. Ludwig, F. Mussnig, Minkowski valuations on convex functions. Calc. Var. Partial Differential Equations56 (2017), Art. 162, 29. MR3715395 Zbl 0682480710.1007/s00526-017-1243-4Search in Google Scholar PubMed PubMed Central

[11] H. Hadwiger, Vorlesungen über Inhalt, Oberfläche und Isoperimetrie. Springer 1957. MR0102775 Zbl 0078.3570310.1007/978-3-642-94702-5Search in Google Scholar

[12] B. J. Kazarnovskiĭ, On zeros of exponential sums. (Russian) Dokl. Akad. Nauk SSSR257 (1981), 804–808. English translation: Sov. Math. Dokl. 23 (1981), 347–351. MR612571 Zbl 0491.3200Search in Google Scholar

[13] B. Y. Kazarnovskiĭ, Newton polyhedra and roots of systems of exponential sums. (Russian) Funktsional. Anal. i Prilozhen. 18 (1984), 40–49, 96. English translation: Functional Anal. Appl. 18 (1984), no. 4, 299–307. MR775932 Zbl 0593.3200110.1007/BF01083691Search in Google Scholar

[14] H. Kone, Valuations on Orlicz spaces and Lϕ-star sets. Adv. in Appl. Math. 52 (2014), 82–98. MR3131927 Zbl 1301.4600910.1016/j.aam.2013.07.004Search in Google Scholar

[15] M. Ludwig, Valuations on function spaces. Adv. Geom. 11 (2011), 745–756. MR2852930 Zbl 1235.5202310.1515/advgeom.2011.039Search in Google Scholar

[16] M. Ludwig, Valuations on Sobolev spaces. Amer. J. Math. 134 (2012), 827–842. MR2931225 Zbl 1255.5201310.1353/ajm.2012.0019Search in Google Scholar

[17] P. McMullen, Valuations and Euler-type relations on certain classes of convex polytopes. Proc. London Math. Soc. (3)35 (1977), 113–135. MR0448239 Zbl 0353.5200110.1112/plms/s3-35.1.113Search in Google Scholar

[18] P. McMullen, Valuations and dissections. In: Handbook of convex geometry, Vol. A, B, 933–988, North-Holland 1993. MR1243000 Zbl 0791.5201410.1016/B978-0-444-89597-4.50010-XSearch in Google Scholar

[19] P. McMullen, R. Schneider, Valuations on convex bodies. In: Convexity and its applications, 170–247, Birkhäuser 1983. MR731112 Zbl 0534.5200110.1007/978-3-0348-5858-8_9Search in Google Scholar

[20] R. Schneider, Convex bodies: the Brunn–Minkowski theory, volume 151 of Encyclopedia of Mathematics and its Applications. Cambridge Univ. Press 2014. MR3155183 Zbl 1287.52001Search in Google Scholar

[21] A. Tsang, Valuations on Lp-spaces. Int. Math. Res. Not. 2010, issue 20, 3993–4023. MR2738348 Zbl 1211.5201310.1093/imrn/rnq028Search in Google Scholar

[22] T. Wang, Semi-valuations on BV(ℝn). Indiana Univ. Math. J. 63 (2014), 1447–1465. MR3283557 Zbl 1320.4603510.1512/iumj.2014.63.5365Search in Google Scholar

Received: 2017-04-26
Revised: 2017-06-15
Published Online: 2019-07-12
Published in Print: 2019-07-26

© 2019 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. A remark on the mixed scalar curvature of a manifold with two orthogonal totally umbilical distributions
  3. On mixed Lp John ellipsoids
  4. Valuations on convex functions and convex sets and Monge–Ampère operators
  5. Weierstrass points on Kummer extensions
  6. The Bonnet problem for harmonic maps to the three-sphere
  7. Maximal arcs in projective planes of order 16 and related designs
  8. On the decomposition of the small diagonal of a K3 surface
  9. Doubly transitive dimensional dual hyperovals: universal covers and non-bilinear examples
  10. Higgs bundles and fundamental group schemes
  11. A relation between the curvature ellipse and the curvature parabola
  12. Non-orientable three-submanifolds of G2-manifolds
  13. The growth of the first non-Euclidean filling volume function of the quaternionic Heisenberg group
  14. A class of analytic pairs of conjugate functions in dimension three
https://doi.org/10.1515/advgeom-2018-0031
Keywords for this article
Valuation on convex bodies; convex set; Monge–Ampère operator

Articles in the same Issue

  1. Frontmatter
  2. A remark on the mixed scalar curvature of a manifold with two orthogonal totally umbilical distributions
  3. On mixed Lp John ellipsoids
  4. Valuations on convex functions and convex sets and Monge–Ampère operators
  5. Weierstrass points on Kummer extensions
  6. The Bonnet problem for harmonic maps to the three-sphere
  7. Maximal arcs in projective planes of order 16 and related designs
  8. On the decomposition of the small diagonal of a K3 surface
  9. Doubly transitive dimensional dual hyperovals: universal covers and non-bilinear examples
  10. Higgs bundles and fundamental group schemes
  11. A relation between the curvature ellipse and the curvature parabola
  12. Non-orientable three-submanifolds of G2-manifolds
  13. The growth of the first non-Euclidean filling volume function of the quaternionic Heisenberg group
  14. A class of analytic pairs of conjugate functions in dimension three
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