## Abstract

Let f : ℝ^{n} → ℝ_{+} be a log-concave function such that (x_{1}, . . . , x_{n}) = f (|x_{1}|, . . . , |x_{n}|) for every (x_{1}, . . . .,x_{n}) ∈ ℝ^{n} and o < ∫_{ℝn} < + ∞.. It was proved in [4] and [5] iffor y ∈ ℝ^{n},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 ≥ 4^{n}, We characterize here the case of equality: one has P(f) = 4^{n}4^{n} if and only if can be written as f(x_{1}, x_{2}) e^{-||x1||K1} 1K2 (x2), where and K_{1}, ⊂ ℝ^{n1} and K_{2} ⊂ ℝ^{n2}, n_{1} + n_{2} = n, are unconditional convex bodies such that P(K_{i}): = |K_{i} |K_{i}^{*}| = 4^{n}i/n _{i}!,i = 1, 2, where K_{i}^{*} denotes the polar of K_{i}. These last bodies were characterized in [7] and [8]

Original language | English |
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Pages (from-to) | 621-630 |

Number of pages | 10 |

Journal | Advances in Geometry |

Volume | 10 |

Issue number | 4 |

DOIs | |

State | Published - Oct 2010 |

## Keywords

- Inverse Santal inequality
- Legendre transform
- Log-concave functions
- Mahler conjecture

## ASJC Scopus subject areas

- Geometry and Topology