A Sharp Bound on Large Planar Signed Vector Sums

TL;DR

This paper establishes a precise lower bound on the maximum Euclidean norm of signed sums of planar vectors, linking it to polygon circumradius and extending results to Minkowski sums of convex bodies in any dimension.

Contribution

It introduces a sharp bound for signed vector sums in the plane and applies it to derive bounds on the circumradius of Minkowski sums of convex bodies.

Findings

Sharp lower bound for signed vector sums in the plane

Tight bounds for Minkowski sums of convex bodies in any dimension

Connection between signed sums and polygon isoperimetric problems

Abstract

We give a sharp lower bound to the largest possible Euclidean norm of signed sums of $n$ vectors in the plane. This is achieved by connecting the signed vector sum problem to the isoperimetric problem for the circumradius of polygons. In turn, we apply the sharp bound for the signed vector sum problem to establish a sharp lower bound to the circumradius of the Minkowski sum of $n$ planar symmetric convex bodies. We also determine a tight lower bound to the circumradius of the Minkowski sum of general convex bodies in any dimension independent of their number.