TY - JOUR
T1 - The santaló point of a function, and a functional form of the santaló inequality
AU - Artstein-Avidan, S.
AU - Klartag, B.
AU - Milman, V.
N1 - Funding Information:
Acknowledgements. The first and third authors were partially supported by a BSF grant. The second author was supported by NSF grant DMS-0111298 and the Bell Companies Fellowship.
PY - 2004
Y1 - 2004
N2 - Let ℒ(f) denote the Legendre transform of a function f: ℝn → ℝ. A theorem of K. Ball about even functions is generalized, and it is proved that, for any measurable function f ≥ 0, there exists a translation f̃(x) = f(x - a) such that ∫ℝne -f̃ ∫ℝne-ℒ(f̃)≤ (2π)n. (1) If a is selected so as to minimize the left side of (1), then equality in (1) is satisfied if and only if e-f is proportional to the distribution of a Gaussian random variable. This inequality immediately implies the Santaló inequality for convex bodies, as well as a new concentration inequality for the Gaussian measure.
AB - Let ℒ(f) denote the Legendre transform of a function f: ℝn → ℝ. A theorem of K. Ball about even functions is generalized, and it is proved that, for any measurable function f ≥ 0, there exists a translation f̃(x) = f(x - a) such that ∫ℝne -f̃ ∫ℝne-ℒ(f̃)≤ (2π)n. (1) If a is selected so as to minimize the left side of (1), then equality in (1) is satisfied if and only if e-f is proportional to the distribution of a Gaussian random variable. This inequality immediately implies the Santaló inequality for convex bodies, as well as a new concentration inequality for the Gaussian measure.
UR - http://www.scopus.com/inward/record.url?scp=33646778163&partnerID=8YFLogxK
U2 - 10.1112/S0025579300015497
DO - 10.1112/S0025579300015497
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AN - SCOPUS:33646778163
SN - 0025-5793
VL - 51
SP - 33
EP - 48
JO - Mathematika
JF - Mathematika
IS - 1-2
ER -