Abstract
Let \(\phi: {\mathbb{R}}^n\to {\mathbb{R}}\cup\{+\infty\}\) be a convex function and \(\mathcal{L}\phi\) be its Legendre tranform. It is proved that if \(\phi\) is invariant by changes of signs, then \(\int e^{-\phi}\int e^{-\mathcal{L}\phi} \ge 4^n\). This is a functional version of the inverse Santaló inequality for unconditional convex bodies due to J. Saint Raymond. The proof involves a general result on increasing functions on \(\mathbb{R}^{n} \times \mathbb{R}^n\) together with a functional form of Lozanovskii’s lemma. In the last section, we prove that for some c > 0, one has always \(\int e^{-\phi}\int e^{-\mathcal{L}\phi} \ge c^n\). This generalizes a result of B. Klartag and V. Milman.
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References
S. Artstein, M. Klartag, V. Milman, The Santaló point of a function and a functional form of Santaló inequality, Mathematika, 51 (2004), 33–48.
K. Ball, Isometric problems in ℓ p and sections of convex sets, Ph.D Dissertation, Cambridge (1986).
W. Blaschke, Über affine Geometrie 7: Neue Extremeigenschaften von Ellipse und Ellipsoid, Wilhelm Blaschke Gesammelte Werke 3, Thales Verlag, Essen (1985), pp. 246–258.
B. Bollobás, I. Leader, Products of unconditional bodies, Oper. Theory Adv. Appl., 77 (1995), 13–24.
B. Bollobás, I. Leader, A.J. Radcliffe, Reverse Kleitman inequalities, Proc. Lond. Math. Soc. 58 (3) (1989), 153–168.
J. Bourgain, V.D. Milman, New volume ratio properties for convex symmetric bodies in R n, Invent. Math., 88(2) (1987), 319–340.
M. Fradelizi, Sections of convex bodies through their centroid, Arch. Math., 69(6) (1997), 515–522.
M. Fradelizi, M. Meyer, Some functional forms of Blaschke-Santaló inequality, Math. Z., 256 (2007), 379–395.
M. Fradelizi, M. Meyer, Some functional form of inverse Santaló inequality (in preparation).
B. Klartag, V. Milman, Geometry of log-concave functions and measures, Geom. Dedic., 112 (2005), 169–182.
G. Ya. Lozanovskii, On some Banach lattices, Siberian, Math. J., 10 (1969), 419–430.
K. Mahler, Ein Übertragungsprinzip für konvexe Körper, Časopis Pěst. Mat. Fys., 68 (1939), 93–102.
M. Meyer, Une caractérisation volumique de certains espaces normés de dimension finie, Isr. J. Math., 55(3) (1986), 317–326.
J. Saint Raymond, Sur le volume des corps convexes symétriques. Séminaire d’Initiation à l’Analyse, 1980/1981, Publ. Math. Univ. Pierre et Marie Curie, Paris (1981).
S. Reisner, Minimal volume product in Banach spaces with a 1-unconditional basis, J. Lond. Math. Soc., 36 (1987), 126–136.
L.A. Santaló, An affine invariant for convex bodies of n-dimensional space. Port. Math. 8, (1949), 155–161.
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Fradelizi, M., Meyer, M. Increasing functions and inverse Santaló inequality for unconditional functions. Positivity 12, 407–420 (2008). https://doi.org/10.1007/s11117-007-2145-z
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DOI: https://doi.org/10.1007/s11117-007-2145-z