  Using a natural representation of a $1/s$-concave function on $\mathbb{R}^d$
as a convex set in $\mathbb{R}^{d+1},$ 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 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 Santal\'o regions for $s$-concave and
log-concave functions and generalize the Santal\'o inequality for them in the
case the origin is not the Santal\'o point.
