Abstract

Let n≥2n\ge2 and let Φ ⁣:Rn→[0,∞)\Phi\colon{\mathbb R}^n\to[0,\infty) be a positively 11-homogeneous and convex function. Given two convex bodies A⊂BA\subset B in Rn{\mathbb R}^n, the monotonicity of anisotropic Φ\Phi-perimeters holds, i.e. PΦ(A)≤PΦ(B)P_\Phi(A)\le P_\Phi(B). In this note, we prove a quantitative lower bound on the difference of the Φ\Phi-perimeters of AA and BB in terms of their Hausdorff distance.

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Keywords

  • Convex body
  • anisotropic perimeter
  • Hausdorff distance
  • Wulff inequality

Mathematics Subject Classification

52A20; 52A40

References

  1. Archimedes: The works of Archimedes. Two Books on the Sphere and the Cylinder, Vol. I; translated into English by R. Netz, Cambridge University Press, Cambridge (2004).Google Scholar
  2. E. Baer: Minimizers of anisotropic surface tensions under gravity: higher dimensions via symmetrization, Arch. Ration. Mech. Anal. 215(2) (2015) 531–578.https://doi.org/10.1007/s00205-014-0788-zGoogle Scholar
  3. T. Bonnesen, W. Fenchel: Theory of Convex Bodies; translated into English by L. Boron, C. Christenson and B. Smith, BCS Associates, Moscow USA (1987).Google Scholar
  4. G. Buttazzo, V. Ferone, B. Kawohl: Minimum problems over sets of concave functions and related questions, Math. Nachr. 173 (1995) 71–89.https://doi.org/10.1002/mana.19951730106Google Scholar
  5. M. Carozza, F. Giannetti, F. Leonetti, A. Passarelli di Napoli: A sharp quantitative estimate for the perimeters of convex sets in the plane, J. Convex Analysis 22(3) (2015) 853–858.https://doi.org/10.68381/jca22043Google Scholar
  6. M. Carozza, F. Giannetti, F. Leonetti, A. Passarelli di Napoli: A sharp quantitative estimate for the surface areas of convex sets in R3, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 27(3) (2016) 327–333.https://doi.org/10.4171/rlm/737Google Scholar
  7. P. M. Gruber: Convex and Discrete Geometry, Grundlehren der Math. Wissenschaften 336, Springer, Berlin (2007).Google Scholar
  8. M. La Civita, F. Leonetti: Convex components of a set and the measure of its boundary, Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia 56 (2008/09) 71–78.Google Scholar
  9. F. Maggi: Sets of Finite Perimeter and Geometric Variational Problems, Cambridge Studies in Advanced Mathematics 135, Cambridge University Press, Cambridge (2012).https://doi.org/10.1017/CBO9781139108133Google Scholar

Suggested citation

G. Stefani. “On the Monotonicity of Perimeter of Convex Bodies.” Journal of Convex Analysis 25 (2018), No. 1, 93–102. https://doi.org/10.68381/jca25006

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Published by Heldermann Verlag, 2018. Rights now held by Banach Press LLC.