On functional successive minima

Abstract

In the classical Geometry of Numbers, the calculation of successive minima may be quite difficult, even in $R^2$ using the norm coming from a distance function associated to a set. In the literature, there seem to be hardly any analogues when $R$ is replaced by the algebraic closure of a function field in one variable and one uses a norm arising from the absolute height. Here, we calculate a one-parameter family of examples that naturally arose in our recent paper on bounded heights. We also comment on whether the minima are attained.