Given n reference points in real d-space, we specify a finite set of hyperplanes that are perpendicular to lines that join pairs of the n points. These hyperplanes dissect the space into a number of regions which is determined by the intersection semilattice of the hyperplanes. The semilattice in turn is, for generic reference points, determined by d and the lift matroid of a gain graph that corresponds to the specifications of the hyperplanes.

Examples include the ``braid arrangements'' and their affine deformations, that have lately attracted interest in some quarters.

Dissections of this kind arise from generalizing a problem in geometric voting theory. I will discuss some particular examples of possible interest for voting.

The talk will to a great extent depend on pictures and will not assume any knowledge of weird technical machinery.