Successive Minima

In this chapter, we define the notion of successive minima of a convex body \(\mathcal {K}\) with respect to a lattice and formulate Minkowski’s Second Theorem for successive minima. We shall give two proofs of Minkowski’s Second Theorem, one due to Bambah–Davenport and another by Danicic. We state Davenport’s Conjecture (which is still open for \(n\ge 4\) ) for product of successive minima of a convex body \(\mathcal {K}\) and give a proof of it when \(n=2\) or when \(\mathcal {K}\) is a sphere. Some related results are also proved. To make the concepts clear and independent, we give detailed proofs of some lemmas which are required to derive these results.