  Let $X$ be a projective variety of dimension $n$ over an algebraically closed
field of arbitrary characteristic and let $A, B, C$ be nef divisors on $X$. We
show that for any integer $1\leq k\leq n-1$, $$ (B^k\cdot A^{n-k})\cdot
(A^k\cdot C^{n-k})\geq \frac{k!(n-k)!}{n!}(A^n)\cdot (B^k\cdot C^{n-k}). $$ The
same inequality in the analytic setting was obtained by Lehmann and Xiao for
compact K\"ahler manifolds using the Calabi--Yau theorem, while our approach is
purely algebraic using (multipoint) Okounkov bodies. We also discuss
applications of this inequality to B\'ezout-type inequalities and inequalities
on degrees of dominant rational self-maps.
