Applications of an Idea of Voronoĭ, a Report

The idea of Voronoĭ’s proof of his well-known criterion that a positive definite quadratic form is extreme if and only if it is eutactic and perfect, is as follows: Identify positive definite quadratic forms on ??

d

with their coefficient vectors in

<math display='block'> <mrow> <msup> <mi mathvariant='double-struck'>E</mi> <mrow> <mfrac> <mn>1</mn> <mi>d</mi> </mfrac> <mrow><mo>(</mo> <mrow> <mi>d</mi><mo>+</mo><mn>1</mn> </mrow> <mo>)</mo></mrow> </mrow> </msup> </mrow> </math>

$${\mathbb{E}^{\frac{1}{d}\left( {d + 1} \right)}}$$

. This translates certain problems on quadratic forms into more transparent geometric problems in

<math display='block'> <mrow> <msup> <mi mathvariant='double-struck'>E</mi> <mrow> <mfrac> <mn>1</mn> <mi>d</mi> </mfrac> <mrow><mo>(</mo> <mrow> <mi>d</mi><mo>+</mo><mn>1</mn> </mrow> <mo>)</mo></mrow> </mrow> </msup> </mrow> </math>

$${\mathbb{E}^{\frac{1}{d}\left( {d + 1} \right)}}$$

which, sometimes, are easier to solve. Since the 1960s this idea has been applied successfully to various problems of quadratic forms, lattice packing and covering of balls, the Epstein zeta function, closed geodesics on the Riemannian manifolds of a Teichmüller space, and other problems.

This report deals with recent applications of Voronoĭ’s idea. It begins with geometric properties of the convex cone of positive definite quadratic forms and a finiteness theorem. Then we describe applications to lattice packings of balls and smooth convex bodies, to the Epstein zeta function and a generalization of it and, finally, to John type and minimum position problems.