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  • Christos Saroglou (a1)


We prove that the log-Brunn–Minkowski inequality (log-BMI) for the Lebesgue measure in dimension $n$ would imply the log-BMI and, therefore, the B-conjecture for any even log-concave measure in dimension $n$ . As a consequence, we prove the log-BMI and the B-conjecture for any even log-concave measure in the plane. Moreover, we prove that the log-BMI reduces to the following: for each dimension $n$ , there is a density $f_{n}$ , which satisfies an integrability assumption, so that the log-BMI holds for parallelepipeds with parallel facets, for the density $f_{n}$ . As a byproduct of our methods, we study possible log-concavity of the function $t\mapsto |(K+_{p}\cdot ~\text{e}^{t}L)^{\circ }|$ , where $p\geqslant 1$ and $K$ , $L$ are symmetric convex bodies, which we are able to prove in some instances and, as a further application, we confirm the variance conjecture in a special class of convex bodies. Finally, we establish a non-trivial dual form of the log-BMI.



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1. Anttila, M., Ball, K. and Perissinaki, I., The central limit problem for convex bodies. Trans. Amer. Math. Soc. 355 2003, 47234735.
2. Ball, K., Some remarks on the geometry of convex sets. In Geometric Aspects of Functional Analysis (1986/87) (Lecture Notes in Mathematics 1317 ), Springer (Berlin, 1988), 224231.
3. Bobkov, S. and Koldobsky, A., On the central limit properties of convex bodies (Lecture Notes in Mathematics 1807 ), Springer (Berlin, 2003), 4452.
4. Borell, C., Convex measures on locally convex spaces. Ark. Mat. 12 1974, 239252.
5. Böröczky, K. J., Lutwak, E., Yang, D. and Zhang, G., The log-Brunn–Minkowski inequality. Adv. Math. 231 2012, 19741997.
6. Böröczky, K. J., Lutwak, E., Yang, D. and Zhang, G., The logarithmic Minkowski problem. J. Amer. Math. Soc. 26 2013, 831852.
7. Böröczky, K. J., Lutwak, E., Yang, D. and Zhang, G., Affine images of isotropic measures. J. Differential Geom. accepted.
8. Bourgain, J., On the distribution of polynomials on high-dimensional convex sets. In Geometric aspects of functional analysis (1989–90) (Lecture Notes in Mathematics 1469 ), Springer (Berlin, 1991), 127137.
9. Colesanti, A. and Fragalà, I., The first variation of the total mass of log-concave functions and related inequalities. Adv. Math. 244 2013, 708749.
10. Cordero-Erausquin, D., Fradelizi, M. and Maurey, B., The (B) conjecture for the Gaussian measure of dilates of symmetric convex sets and related problems. J. Funct. Anal. 214 2004, 410427.
11. Cordero-Erausquin, D. and Gozlan, N., Transport proofs of weighted Poincaré inequalities for log-concave distributions. Preprint, 2014, arXiv:1407.3217.
12. Eldan, R., Thin shell implies spectral gap via a stochastic localization scheme. Geom. Funct. Anal. 23(2) 2013, 532569.
13. Eldan, R. and Klartag, B., Approximately gaussian marginals and the hyperplane conjecture. In Proceedings of a workshop on “Concentration, Functional Inequalities and Isoperimetry” (Contemporary Mathematics 545 ), American Mathematical Society (Providence, RI, 2011), 5568.
14. Firey, W. J., Polar means of convex bodies and a dual to the Brunn–Minkowski theorem. Canad. J. Math. 13 1961, 444453.
15. Firey, W. J., Mean cross-section measures of harmonic means of convex bodies. Pacific J. Math. 11 1961, 12631266.
16. Fleury, B., Between Paouris concentration inequality and variance conjecture. Ann. Inst. Henri Poincaré Probab. Stat. 46(2) 2010, 299312.
17. Fradelizi, M., Guédon, O. and Pajor, A., Spherical thin-shell concentration for convex measures. Preprint.
18. Gardner, R. J., Geometric Tomography, 2nd edn., Cambridge University Press (Cambridge, 2006).
19. Gardner, R. J., The Brunn–Minkowski inequality. Bull. Amer. Math. Soc. 39 2002, 355405.
20. Gardner, R. J. and Zvavitch, A., Gaussian Brunn–Minkowski inequalities. Trans. Amer. Math. Soc. 362(10) 2010, 53335353.
21. Gardner, R. J., Hug, D., Weil, W. and Ye, D., The dual Orlicz-Brunn–Minkowski theory. Preprint, 2014.
22. Guédon, O. and Milman, E., Interpolating thin-shell and sharp large-deviation estimates for isotropic log-concave measures. Geom. Funct. Anal. 21(5) 2011, 10431068.
23. Guédon, O., Concentration phenomena in high dimensional geometry. In ESAIM Proceedings, Vol. 44 (SMAI) (2014), 4760.
24. Hernández Cifre, M. A. and Yepes Nicolás, J., On Brunn–Minkowski type inequalities for polar bodies. J. Geom. Anal. to appear.
25. Kannan, R., Lovász, L. and Simonovits, M., Isoperimetric problems for convex bodies and a localization lemma. Discrete Comput. Geom. 13(3–4) 1995, 541559.
26. Klartag, B., On convex perturbations with a bounded isotropic constant. Geom. Funct. Anal. 16(6) 2006, 12741290.
27. Klartag, B., A Berry-Esseen type inequality for convex bodies with an unconditional basis. Probab. Theory Related Fields 45(1) 2009, 133.
28. Klartag, B., High-dimensional distributions with convexity properties. In Proceedings of the Fifth European Congress of Mathematics, Amsterdam, July 2008, European Mathematical Society publishing house (2010), 401417.
29. Latala, R. and Oleszkiewicz, K., Small ball probability estimate in terms of width. Studia Math. 169 2005, 305314.
30. Latala, R., On some inequalities for Gaussian measures. In Proceedings of the International Congress of Mathematicians, Beijing, Vol. II, Higher Education Press (Beijing, 2002), 813822.
31. Livne Bar-on, A., The (B) conjecture for uniform measures in the plane. Preprint, 2013, arXiv:1311.6584.
32. Lutwak, E., The Brunn–Minkowski–Firey theory. I. Mixed volumes and the Minkowski problem. J. Differential Geom. 38 1993, 131150.
33. Lutwak, E., The Brunn–Minkowski–Firey theory. II. Affine and geominimal surface areas. Adv. Math. 118 1996, 224294.
34. Lutwak, E., Yang, D. and Zhang, G., L p affine isoperimetric inequalities. J. Differential Geom. 56 2000, 111132.
35. Lutwak, E., Yang, D. and Zhang, G., On the L p -Minkowski problem. Trans. Amer. Math. Soc. 356 2004, 43594370.
36. Lutwak, E., Yang, D. and Zhang, G., L p John Ellipsoids. Proc. Lond. Math. Soc. 90 2005, 497520.
37. Mahler, K., Ein Minimalproblem für konvexe Polygone. Mathematica (Zutphen) B 7 1939, 118127.
38. Marsiglietti, A., On the improvement of concavity of convex measures. Preprint, 2014, arXiv:1403.7643.
39. Milman, V. D. and Pajor, A., Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space. In Geometric Aspects of Functional Analysis (1987–88) (Lecture Notes in Mathematics 1376 ), Springer (Berlin, 1989), 64104.
40. Saroglou, C., Shadow systems: remarks and extensions. Arch. Math. 100 2013, 389399.
41. Saroglou, C., Remarks on the conjectured log-Brunn–Minkowski inequality. Geom. Dedicata 2013, to appear.
42. Schneider, R., Convex Bodies: The Brunn–Minkowski Theory, 2nd edn., Cambridge University Press (Cambridge, 2014).
43. Stancu, A., The discrete planar L 0 -Minkowski problem. Adv. Math. 167 2002, 160174.
44. Stancu, A., On the number of solutions to the discrete two-dimensional L 0 -Minkowski problem. Adv. Math. 180 2003, 290323.
45. Uhrin, B., Curvilinear extensions of the Brunn–Minkowski–Lusternik inequality. Adv. Math. 109(2) 1994, 288312.
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