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The case of equality for an inverse Santaló functional inequality
-
M. Fradelizi
Published/Copyright:
April 23, 2010
Abstract
Let 
be a log-concave function such that ƒ(x1, . . . , xn) = ƒ(|x1|, . . . , |xn|) for every 
and 
. It was proved in [Fradelizi, Meyer, Positivity 12: 407–420, 2008] and [Fradelizi, Meyer, Adv. Math. 218: 1430–1452, 2008] that if for 
, 
, then
We characterize here the case of equality: one has P(ƒ) = 4n if and only if ƒ can be written as
ƒ (x1, x2) = e−‖x1‖K11K2 (x2),
where 
and 
, n1 + n2 = n, are unconditional convex bodies such that 
, i = 1, 2, where 
denotes the polar of Ki. These last bodies were characterized in [Meyer, Israel J. Math. 55: 317–326, 1986] and [Reisner, J. London Math. Soc. 36: 126–136, 1987].
Key words.: Log-concave functions; Legendre transform; Mahler conjecture; Inverse Santaló inequality
Received: 2008-05-09
Published Online: 2010-04-23
Published in Print: 2010-October
© de Gruyter 2010
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Keywords for this article
Log-concave functions;
Legendre transform;
Mahler conjecture;
Inverse Santaló inequality
Articles in the same Issue
- A combinatorial approach to Alexander–Hirschowitz's Theorem based on toric degenerations
- Triply periodic minimal surfaces which converge to the Hoffman–Wohlgemuth example
- Total absolute horospherical curvature of submanifolds in hyperbolic space
- The case of equality for an inverse Santaló functional inequality
- Circle configurations in strictly convex normed planes
- Rank four vector bundles without theta divisor over a curve of genus two
- Affine Tallini Sets and Grassmannians
- Geometry of self-dual flats over a PID on a polarity
- Logarithmic comparison theorem versus Gauss–Manin system for isolated singularities
- A kinematic formula for the total absolute curvature of intersections
- Reducible Veronese surfaces
- Effective non-vanishing conjectures for projective threefolds
