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Let $K$ be an algebraic number field and ${𝒪}_K$ the ring of integers of $K$. In this paper, we prove an analogue of Voronoï’s theorem for ${𝒪}_K$-lattices and the finiteness of the number of similar isometry classes of perfect ${𝒪}_K$-lattices.

We generalize Poor and Yuen’s inequality to the Hermite–Rankin constant ${\gamma }_{n,k}$ and the Bergé–Martinet constant ${\gamma }_{n,k}^{\prime }$. Moreover, we determine explicit values of some low- dimensional Hermite–Rankin and Bergé–Martinet constants by applying Rankin’s inequality and some inequalities proven by Bergé and Martinet to explicit values of ${\gamma }_5^{\prime },{\gamma }_7^{\prime }$, ${\gamma }_{4,2}$ and ${\gamma }_n$ ($n\leqq 8$).