Extremal affine surface areas in a functional setting

We introduce extremal affine surface areas in a functional setting. We show their main properties. Among them are linear invariance, isoperimetric inequalities and monotonicity properties. We establish a new duality formula, which shows that the maximal (resp. minimal) inner affine surface area of an s-concave function on Rⁿ equals the maximal (resp. minimal) outer affine surface area of its Legendre polar. We estimate the ``size" of these quantities: up to a constant depending on n and s only, the extremal affine surface areas are proportional to a power of the integral of f. This extends results obtained in the setting of convex bodies. We recover and improve those as a corollary to our results.