Characterization of products of projective spaces via nef complexity

We define the nef complexity of a projective variety $X$. This invariant compares $\dim X+\rho(X)$ with the sum of the coefficients of nef partitions of $-K_X$. We prove that the nef complexity is non-negative and it is zero precisely for products of projective spaces. We classify smooth Fano threefolds with nef complexity at most one. In a similar vein, we prove Mukai's conjecture for smooth Fano varieties for which every extremal contraction is of fiber type and study smooth images of products of projective spaces. Along the way, we answer positively a question of J. Starr regarding the nef cone of smooth Fano varieties.