Abstract

It is known that the volume functional  ϕ↦∫e−ϕ \,\phi\mapsto\int e^{-\phi}\, satisfies certain concavity or convexity inequalities with respect to three of the four linear structures induced by the order isomorphisms acting on Cvx0(Rn){\rm{Cvx}}_0(\mathbb{R}^n). In this note we define the fourth linear structure on Cvx0(Rn){\rm{Cvx}}_0(\mathbb{R}^n) as the pullback of the standard linear structure under the J\cal J transform. We show that, interpolating with respect to this linear structure, no concavity or convexity inequalities hold, and prove that a quasi-convexity inequality is violated only by up to a factor of 22. We also establish all the order relations which the four different interpolations satisfy.

Author information

Contact details are reproduced from the original publication and may be historical.

Dan I. Florentin

Department of Mathematics, Cleveland State University, Cleveland, OH 44115-2214, U.S.A.

danflorentin@gmail.com

Alexander Segal

Afeka Academic College of Engineering, Tel Aviv 69107, Israel

segalale@gmail.com

Keywords

  • Convexity
  • interpolation
  • order isomorphisms
  • duality
  • Legendre transform
  • A-transform
  • J-transform

Mathematics Subject Classification

26B15, 26B25, 26B35, 39B62, 46B06, 52A23, 52A40, 52A41

References

  1. S. Artstein-Avidan, D. I. Florentin, A. Segal: Functional Brunn-Minkowski inequalities induced by polarity, Advances in Mathematics 364 (2020), https://doi.org/10.1016/j.aim.2020.107006.https://doi.org/10.1016/j.aim.2020.107006Google Scholar
  2. S. Artstein-Avidan, V. D. Milman: Hidden structures in the class of convex functions and a new duality transform, J. European Math. Soc. 13(4) (2011) 975–1004.https://doi.org/10.4171/jems/273Google Scholar
  3. W. J. Firey: Polar means of convex bodies and a dual to the Brunn-Minkowski theorem, Canad. J. Math. 13 (1961) 444–453.https://doi.org/10.4153/CJM-1961-037-0Google Scholar
  4. L. Leindler: On certain converse of Hölder’s inequality II, Acta Scient. Math. (Szeged) 33(3-4) (1972) 217–223.Google Scholar
  5. A. Prékopa: Logarithmic concave measures with application to stochastic programming, Acta Scient. Math. (Szeged) 32 (1971) 301–316.Google Scholar

Suggested citation

D. I. Florentin, A. Segal. “On the Linear Structures Induced by the Four Order Isomorphisms Acting on Cvx_0(ℝ^n).” Journal of Convex Analysis 28 (2021), No. 3, 751–760. https://doi.org/10.68381/jca28042

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2021. Rights now held by Banach Press.