The Santalo point of a function, and a functional form of the Santalo inequality

By Shiri Artstein-Avidan, Bo'az Klartag, V. Milman. 2004

196 citations recorded by the source corpus.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/59002e5d5e2f5a37d6a7bf723be35a95db1f6d12

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Citations in the corpus

Partitions and functional Santaló inequalities (2009, background): “Later on, Artstein, Klartag and Milman [1] proved that any integrable f admits a Santaló point with respect to the weight t 7→ e−t2/2.”
On the infimum convolution inequality (2008, methodology): “There exists a constant C such that for any p ∈ [1, 2], t > 0 and n ∈ N we have ν p ( A+ C(tB p + t B 2 ) ) ≥ min {1 2 , eν p (A) }”
Marginals of Geometric Inequalities (2007, unknown): “Sketch of Proof: By Lemma 3.2 in [ 3 ], we have that L′fk → L′f locally uniformly.”
Some functional forms of Blaschke–Santaló inequality (2006, background): “Proof (A) The inequality We apply Proposition 3 to f1(x) = h(|x|)χK(x), f2(y) = h(|y|)χK◦(y) and ρ(t) = h( √ t)χ[0,1](t).”
Convex bodies and norms associated to convex measures (2010, background): “for any v in Fn with w = Lv. Only recently, in [ 1 ], this inequality was understood as the Santalo-type inequality for the canonical polarity in the class Fn. In modern”
Some functional inverse Santalo inequalities (2008, methodology): “See for instance [1,2,5–8,10,13,14,16].”
Hidden structures in the class of convex functions and a new duality transform (2011, background): “The proof is complete. ut”
A characterization of the concept of duality (2007, background): “The original question, which he posed on several occasions, was regarding the usual polarity of convex bodies in Example 2.”
Rogers–Shephard inequality for log-concave functions (2014, background): “This has generated an increasing interest to extend several important parameters of convex bodies to functional parameters [5,6,10,12,19,22,24] in the class of log-concave functions.”
Moment Measures (2013, background): “(52) Since the log-concave function e−ψ has barycenter at the origin, the functional Santaló inequality from Artstein, Klartag and Milman [2] asserts that ∫ Rn e−ψ ∫ Rn e−ϕ ≤ (2π)n. (53) Now (50) follows from (52) and (53), and the lemma is proven.”
The concept of duality for measure projections of convex bodies (2008, unknown): “Keywords: Duality; Convexity; s-Concavity”
Divergence for s-concave and log concave functions (2013, background): “own aﬃne isoperimetric inequalities is the Blaschke Santal´o inequality [10, 49, 59]. The corresponding inequalities for log concave functions were proved by Ball [7] and Artstein, Klartag and Milman [5] (see also [17, 31]). A stronger inequality than the Blaschke Santal´o inequality is the aﬃne iso”
Real Monge-Ampere equations and Kahler-Ricci solitons on toric log Fano varieties (2012, background): “Interestingly, the boundedness of G under the moment condition on (P, g), say for g = 1, is also a consequence of the functional form of the Santaló inequality [3].”
Mass Transport and Variants of the Logarithmic Sobolev Inequality (2007, background): “ved by a factor 2 for centered functions, whereas the log-Sobolev inequality does not improve for even functions. It was recently understood that the τ-property can be improved for centered functions [50, 6]. The constant 1 2 in the above transportation cost inequality cannot be improved for symmetr”
Quermassintegrals of quasi-concave functions and generalized Prékopa–Leindler inequalities (2012, result): “As a significant example, one may indicate the celebrated Prékopa–Leindler inequality [13,23,28–30] (see also [5,6,9] for recent related papers), or the functional form of the Blaschke–Santaló inequality [2,3].”
Increasing functions and inverse Santaló inequality for unconditional functions (2008, background): “For every z ! Rn, we define # z(x )= # (z + x). As observed in [ 1 ], the function”