The case of equality for an inverse Santal functional inequality

Let f : ℝⁿ → ℝ₊ be a log-concave function such that (x₁, . . . , x_n) = f (|x₁|, . . . , |x_n|) for every (x₁, . . . .,x_n) ∈ ℝⁿ and o < ∫_ℝn < + ∞.. It was proved in [4] and [5] iffor y ∈ ℝⁿ,f^* Fradelizi, Meyer, Positivity 12: 407420, 2008 and Fradelizi(y): = inf _x∈ℝn e^-(x,y)/f(x) then P (f) := ∫_ℝn f(x) dx ∫_ℝn f^*(y) dy ≥ 4ⁿ, We characterize here the case of equality: one has P(f) = 4ⁿ4ⁿ if and only if can be written as f(x₁, x₂) e^-||x1||K1 1K2 (x2), where and K₁, ⊂ ℝⁿ¹ and K₂ ⊂ ℝⁿ², n₁ + n₂ = n, are unconditional convex bodies such that P(K_i): = |K_i |K_i^*| = 4ⁿi/n _i!,i = 1, 2, where K_i^* denotes the polar of K_i. These last bodies were characterized in [7] and [8]