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 $\mathbb{R}^n$ 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.