The Number of Singularities in the Intersections of Convex Planar Translates

TL;DR

This paper proves that the intersection of n translates of a strictly convex, smooth body in the plane has exactly n boundary singularities if it has a non-empty interior and all translates contribute, with the result being sharp.

Contribution

It establishes a precise count of boundary singularities in intersections of convex translates, highlighting the necessity of all assumptions for the result.

Findings

The intersection has exactly n boundary singularities under given conditions.

Removing any assumption invalidates the result.

The result is optimal and cannot be improved.

Abstract

This purpose of this paper is to prove the following result: let phi be a strictly convex, smooth, convex body in the Euclidean plane, if the intersection of n translates of phi has a non-empty interior, and all of the translates contribute to the intersection, then the intersection of these n translates will have exactly n points of singularity along its boundary. Furthermore this result is sharp, in the sense that, removing any one of the assumptions from our statement will render the result unable to hold in general.