In recent years, the interaction between the local positivity of divisors and Okounkov bodies has attracted considerable attention, and there have been attempts to find a satisfactory theory of positivity of divisors in terms of convex geometry of Okounkov bodies. Many interesting results in this direction have been established by Choi--Hyun--Park--Won and K\"{u}ronya--Lozovanu separately. The first aim of this paper is to give uniform proofs of these results. Our approach provides not only a simple new outlook on the theory but also proofs for positive characteristic in the most important cases. Furthermore, we extend the theorems on Seshadri constants to graded linear series setting. Finally, we introduce the integrated volume function to investigate the relation between Seshadri constants and filtered Okounkov bodies introduced by Boucksom--Chen.

中文翻译：

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重新审视 Seshadri 常数和 Okounkov 体

近年来，约数的局部正性与奥孔科夫体的相互作用引起了相当多的关注，并试图从奥孔科夫体的凸几何方面寻找一个令人满意的约数正性理论。Choi--Hyun--Park--Won 和 K\"{u}ronya--Lozovanu 分别在这个方向建立了许多有趣的结果。本文的第一个目的是给出这些结果的统一证明。我们的方法不仅为理论提供了一个简单的新观点，而且在最重要的情况下也证明了正特性。此外，我们将 Seshadri 常数的定理扩展到分级线性级数设置。最后，