On the constant in the reverse Brunn-Minkowski inequality for p-convex balls
نویسنده
چکیده
This note is devoted to the study of the dependence on p of the constant in the reverse Brunn-Minkowski inequality for p-convex balls (i.e. p-convex symmetric bodies). We will show that this constant is estimated as c ≤ C(p) ≤ C , for absolute constants c > 1 and C > 1. Let K ⊂ IR n and 0 < p ≤ 1. K is called a p-convex set if for any λ, μ ∈ (0, 1) such that λ + μ = 1 and for any points x, y ∈ K the point λx+ μy belongs to K. We will call a p-convex compact centrally symmetric body a p-ball. Recall that a p-norm on real vector space X is a map ‖ · ‖ : X −→ IR such that 1) ‖x‖ > 0 ∀x 6= 0, 2) ‖tx‖ = |t| ‖x‖ ∀t ∈ R, x ∈ X, 3) ∀x, y ∈ X ‖x+ y‖ ≤ ‖x‖ + ‖y‖ . Note that the unit ball of p-normed space is a p-ball and, vice versa, the gauge of p-ball is a p-norm. Recently, J. Bastero, J. Bernués, and A. Peña ([BBP]) extended the reverse Brunn-Minkowski inequality, which was discovered by V. Milman ([M]), to the class of p-convex balls. They proved the following theorem. Theorem Let 0 < p ≤ 1. There exists a constant C = C(p) ≥ 1 such ∗This research was supported by Grant No.92-00285 from United States-Israel Binational Science Foundation (BSF).
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