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Given a nef and big line bundle $L$ on a projective variety $X$ of dimension$d \geq 2$, we prove that the Seshadri constant of $L$ at a very general pointis larger than $(d+1)^{\frac{1}{d}-1}$. This slightly improves the lower bound$1/d$ established by Ein, K\"uchle and Lazarsfeld. The proof relies on theconcept of successive minima for line bundles recently introduced by Ambro andIto.

Lower bounds for Seshadri constants via successive minima of line bundles