Via multilinear algebra, we formulate a criterion for connectedness in the
parametric geometry of numbers in terms of pencils, which are certain algebraic
varieties in the space of matrices. As a consequence, we obtain a connectedness
result for generic lattices arising from Diophantine approximation on analytic
submanifolds, and sharpen Schmidt and Summerer’s results of connectedness
on simultaneous Diophantine approximation and approximation by linear
forms.
