A note on Santaló inequality for the polarity transform and its reverse

We prove a Santaló and a reverse Santaló inequality for the class consisting of even log-concave functions attaining their maximal value 1 at the origin, also called even geometric log-concave functions. We prove that there exist universal numerical constants c,C > 0 such that for any even geometric log-concave function f = e^−ϕ, (equation found) where Bⁿ₂ is the Euclidean unit ball of ℝⁿ and ϕ° is the polar function of ϕ (not the Legendre transform!), a transform which was recently rediscovered by Artstein-Avidan and Milman and is defined below. The bounds are sharp up to the optimal constants c,C.