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.
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© 2014 American Mathematical Society.
- Log-concave function
- Polarity transform
- Santaló and reverse Santaló inequality