Abstract
AbstractUsing a natural representation of a 1/s-concave function on $${\mathbb {R}}^d$$
R
d
as a convex set in $${\mathbb {R}}^{d+1},$$
R
d
+
1
,
we derive a simple formula for the integral of its s-polar. This leads to convexity properties of the integral of the s-polar function with respect to the center of polarity. In particular, we prove that the reciprocal of the integral of the polar function of a log-concave function is log-concave as a function of the center of polarity. Also, we define the Santaló regions for s-concave and log-concave functions and generalize the Santaló inequality for them in the case the origin is not the Santaló point.
Funder
National Science Foundation
Publisher
Springer Science and Business Media LLC
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