Primitive minima of positive definite quadratic forms

The main purpose of the reduction theory is to construct a fundamental domain of the unimodular group acting discontinuously on the space of positive definite quadratic forms. This fundamental domain is for example used in the theory of automorphic forms for GLₙ (cf. [11]) or in the theory of Siegel modular forms (cf. [1], [4]). There are several ways of reduction, which are usually based on various minima of the quadratic form, e.g. the Korkin-Zolotarev method (cf. [10], [3]), Venkov's method (cf. [12]) or Voronoï's approach (cf. [13]), which also works in the general setting of positivity domains (cf. [5]). The most popular method is Minkowski's reduction theory [6] and its generalizations (cf. [9], [15]).
Minkowski's reduction theory is based on attaining certain minima, which can be characterized as the successive primitive minima of the quadratic form. Besides these we have successive minima, but a reduction according to successive minima only works for n ≤ 4 (cf. [14]). In this paper we introduce so-called primitive minima}, which lie between successive and successive primitive minima (cf. Theorem 2). Using primitive minima we obtain a straightforward generalization of Hermite's inequality in Theorem 1. As an application we get a simple proof for the finiteness of the class number. Finally we describe relations with Rankin's minima (cf. [8]) and with Venkov's reduction (cf. [12]).