TY - JOUR
T1 - Geometry of log-concave functions and measures
AU - Klartag, B.
AU - Milman, V. D.
PY - 2005/4
Y1 - 2005/4
N2 - We present a view of log-concave measures, which enables one to build an isomorphic theory for high dimensional log-concave measures, analogous to the corresponding theory for convex bodies. Concepts such as duality and the Minkowski sum are described for log-concave functions. In this context, we interpret the Brunn-Minkowski and the Blaschke-Santaló inequalities and prove the two corresponding reverse inequalities. We also prove an analog of Milman's quotient of subspace theorem, and present a functional version of the Urysohn inequality.
AB - We present a view of log-concave measures, which enables one to build an isomorphic theory for high dimensional log-concave measures, analogous to the corresponding theory for convex bodies. Concepts such as duality and the Minkowski sum are described for log-concave functions. In this context, we interpret the Brunn-Minkowski and the Blaschke-Santaló inequalities and prove the two corresponding reverse inequalities. We also prove an analog of Milman's quotient of subspace theorem, and present a functional version of the Urysohn inequality.
KW - Geometric inequalities
KW - Log-concave functions
KW - Log-concave measures
KW - Reverse Brunn-Minkowski
KW - Reverse Santalò
UR - http://www.scopus.com/inward/record.url?scp=23944445093&partnerID=8YFLogxK
U2 - 10.1007/s10711-004-2462-3
DO - 10.1007/s10711-004-2462-3
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AN - SCOPUS:23944445093
SN - 0046-5755
VL - 112
SP - 169
EP - 182
JO - Geometriae Dedicata
JF - Geometriae Dedicata
IS - 1
ER -