  Let $X$ be a normal and geometrically integral projective variety over a
global field $K$ and let $\overline{D}$ be an adelic Cartier divisor on $X$. We
prove a conjecture of Chen, showing that the essential minimum
$\zeta_{\mathrm{ess}}(\overline{D})$ of $\overline{D}$ equals its asymptotic
maximal slope under mild positivity assumptions. As an application, we see that
$\zeta_{\mathrm{ess}}(\overline{D})$ can be read on the Okounkov body of the
underlying divisor $D$ via the Boucksom--Chen concave transform. This gives a
new interpretation of Zhang's inequalities on successive minima and a criterion
for equality generalizing to arbitrary projective varieties a result of Burgos
Gil, Philippon and Sombra concerning toric metrized divisors on toric
varieties. When applied to a projective space $X = \mathbb{P}_K^d$, our main
result has several applications to the study of successive minima of hermitian
vector spaces. We obtain an absolute transference theorem with a linear upper
bound, answering a question raised by Gaudron. We also give new comparisons
between successive slopes and absolute minima, extending results of Gaudron and
R\'emond.

Algebraic Geometry;Number Theory