Let $X$ be a normal and geometrically integral projective variety over a global field $K$ and let $\bar {D}$ be an adelic ${\mathbb {R}}$-Cartier divisor on $X$. We prove a conjecture of Chen, showing that the essential minimum $\zeta _{\mathrm {ess}}(\bar {D})$ of $\bar {D}$ equals its asymptotic maximal slope under mild positivity assumptions. As an application, we see that $\zeta _{\mathrm {ess}}(\bar {D})$ 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 Rémond.

令$X$是全局域$ K $上的正规和几何积分射影变体，并令$\bar {D}$是$X$上的一个 adelic ${\mathbb {R}}$ -Cartier 除数。我们证明了陈的一个猜想，显示出必要的最低$ \泽塔_ {\ mathrm {ESS}}（\ {条} d）$的$ \ {条} d $在温和的积极性假设等于其渐近最大斜角。作为应用程序，我们看到$\zeta _{\mathrm {ess}}(\bar {D})$可以在底层除数$D$的 Okounkov 体上读取通过 Boucksom-Chen 凹变换。这给出了对连续极小值的 Zhang 不等式的新解释，并给出了将等式推广到任意射影变体的标准，这是 Burgos Gil、Philippon 和 Sombra 关于复曲面变体的复曲面度量化除数的结果。当应用于射影空间$X = {\mathbb {P}}_K^{d}$ 时，我们的主要结果对研究厄密向量空间的连续最小值有多种应用。我们得到了一个具有线性上限的绝对转移定理，回答了 Gaudron 提出的一个问题。我们还给出了连续斜率和绝对最小值之间的新比较，扩展了 Gaudron 和 Rémond 的结果。