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CONCAVITY PROPERTIES OF EXTENSIONS OF THE PARALLEL VOLUME
Published online by Cambridge University Press: 13 January 2015
Abstract
In this paper we establish concavity properties of two extensions of the classical notion of the outer parallel volume. On the one hand, we replace the Lebesgue measure by more general measures. On the other hand, we consider a functional version of the outer parallel sets.
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References
Artstein, S., Klartag, B. and Milman, V., The Santaló point of a function and a functional form of Santaló inequality. Mathematika 51 2004, 33–48.CrossRefGoogle Scholar
Ball, K., Isometric problems in 
$\ell _{p}$ and sections of convex sets. PhD Dissertation, Cambridge, 1986.Google Scholar
$\ell _{p}$ and sections of convex sets. PhD Dissertation, Cambridge, 1986.Google ScholarBobkov, S. G., Isoperimetric and analytic inequalities for log-concave probability measures. Ann. Probab. 27(4) 1999, 1903–1921.CrossRefGoogle Scholar
Bobkov, S. G., Gentil, I. and Ledoux, M., Hypercontractivity of Hamilton–Jacobi equations. J. Math. Pures Appl. (9) 80(7) 2001, 669–696.CrossRefGoogle Scholar
Bobkov, S. G. and Ledoux, M., From Brunn–Minkowski to Brascamp–Lieb and to logarithmic Sobolev inequalities. Geom. Funct. Anal. 10(5) 2000, 1028–1052.CrossRefGoogle Scholar
Bobkov, S. G. and Ledoux, M., Weighted Poincaré-type inequalities for Cauchy and other convex measures. Ann. Probab. 37(2) 2009, 403–427.CrossRefGoogle Scholar
Borell, C., Convex measures on locally convex spaces. Ark. Mat. 12 1974, 239–252.CrossRefGoogle Scholar
Borell, C., Convex set functions in d-space. Period. Math. Hungar. 6 1975, 111–136.CrossRefGoogle Scholar
Borell, C., The Brunn–Minkowski inequality in Gauss space. Invent. Math. 30(2) 1975, 207–216.CrossRefGoogle Scholar
Brascamp, H. J. and Lieb, E. H., On extensions of the Brunn–Minkowski and Prékopa–Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation. J. Funct. Anal. 22 1976, 366–389.CrossRefGoogle Scholar
Cianchi, A., Fusco, N., Maggi, F. and Pratelli, A., On the isoperimetric deficit in Gauss space. Amer. J. Math. 133(1) 2011, 131–186.CrossRefGoogle Scholar
Cordero-Erausquin, D. and Klartag, B., Interpolations, convexity and geometric inequalities. In Geometric Aspects of Functional Analysis (Lecture Notes in Mathematics 2050), Springer (Heidelberg, 2012), 151–168.CrossRefGoogle Scholar
Costa, M., A new entropy power inequality. IEEE Trans. Inform. Theory 31 1985, 751–760.CrossRefGoogle Scholar
Costa, M. and Cover, T. M., On the similarity of the entropy power inequality and the Brunn–Minkowski inequality. IEEE Trans. Inform. Theory 30(6) 1984, 837–839.CrossRefGoogle Scholar
De Castro, Y., Quantitative isoperimetric inequalities on the real line. Ann. Math. Blaise Pascal 18(2) 2011, 251–271.CrossRefGoogle Scholar
Evans, L. C., Partial Differential Equations (Graduate Studies in Mathematics 19), American Mathematical Society (Providence, RI, 1998).Google Scholar
Fradelizi, M., Concentration inequalities for s-concave measures of dilations of Borel sets and applications. Electron. J. Probab. 14(71) 2009, 2068–2090.CrossRefGoogle Scholar
Fradelizi, M. and Guédon, O., The extreme points of subsets of s-concave probabilities and a geometric localization theorem. Discrete Comput. Geom. 31(2) 2004, 327–335.CrossRefGoogle Scholar
Fradelizi, M. and Marsiglietti, A., On the analogue of the concavity of entropy power in the Brunn–Minkowski theory. Adv. Appl. Math. 57 2014, 1–20.CrossRefGoogle Scholar
Fradelizi, M. and Meyer, M., Some functional forms of Blaschke–Santaló inequality. Math. Z. 256(2) 2007, 379–395.CrossRefGoogle Scholar
Fradelizi, M. and Meyer, M., Functional inequalities related to Mahler’s conjecture. Monatsh. Math. 159(1–2) 2010, 13–25.CrossRefGoogle Scholar
Gozlan, N., Roberto, C. and Samson, P.-M., Hamilton Jacobi equations on metric spaces and transport-entropy inequalities. Rev. Mat. Iberoam. 30(1) 2014, 133–163.CrossRefGoogle Scholar
Hardy, G. H., Littlewood, J. E. and Pólya, G., Inequalities, Cambridge University Press (Cambridge, 1959).Google Scholar
Kannan, R., Lovász, L. and Simonovits, M., Isoperimetric problems for convex bodies and a localization lemma. Discrete Comput. Geom. 13(3–4) 1995, 541–559 (English summary).CrossRefGoogle Scholar
Kolesnikov, A. V. and Milman, E., Poincaré and Brunn–Minkowski inequalities on weighted Riemannian manifolds with boundary. Preprint, 2013, arXiv:1310.2526 [math.DG].Google Scholar
Lehec, J., A direct proof of the functional Santaló inequality. C. R. Math. Acad. Sci. Paris 347(1–2) 2009, 55–58.CrossRefGoogle Scholar
Leindler, L., On a certain converse of Hölder’s inequality, II. Acta Sci. Math. (Szeged) 33 1972, 217–223.Google Scholar
Nguyen, V. H., Dimensional variance inequalities of Brascamp–Lieb type and a local approach to dimensional Prékopa’s theorem. J. Funct. Anal. 266(2) 2014, 931–955.CrossRefGoogle Scholar
Prékopa, A., On logarithmic concave measures and functions. Acta Sci. Math. (Szeged) 34 1973, 335–343.Google Scholar
Sudakov, V. N. and Cirel’son, B. S., Extremal properties of half-spaces for spherically invariant measures (Russian) Problems in the theory of probability distributions, II. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 41 1974, 14–24, 165.Google Scholar
Villani, C., A short proof of the “concavity of entropy power”. IEEE Trans. Inform. Theory 46(4) 2000, 1695–1696.CrossRefGoogle Scholar