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A Sharp Bound on Large Planar Signed Vector Sums

Florian Grundbacher

TL;DR

The paper tackles the problem of obtaining sharp lower bounds for the largest Euclidean norm of signed sums of $n$ planar vectors, linking this discrete problem to the isoperimetric problem for the circumradius of polygons. The authors derive an exact value for $c(2,n,n)$ by analyzing the circumradius of zonotopes and applying a Dowker-type polygon inequality, then extend the approach to obtain a tight lower bound on the circumradius of the Minkowski sum of $n$ planar symmetric convex bodies and, more generally, a dimension-independent bound via intrinsic volumes. The main contributions include the precise bound $c(2,n,n)=1/\sin(\pi/(2n))$, a sharp inequality for $R(K^1+\cdots+K^n)$ in terms of $R(K^i)$ for symmetric planar bodies, and a best-possible absolute constant in the general d-dimensional setting, together with discussions of equality cases and potential generalizations to non-symmetric bodies and higher dimensions. The work connects discrete vector-sum problems with convex-geometric isoperimetric questions, yielding tight, widely applicable inequalities with implications for the geometry of zonotopes and Minkowski sums.

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.

A Sharp Bound on Large Planar Signed Vector Sums

TL;DR

The paper tackles the problem of obtaining sharp lower bounds for the largest Euclidean norm of signed sums of planar vectors, linking this discrete problem to the isoperimetric problem for the circumradius of polygons. The authors derive an exact value for by analyzing the circumradius of zonotopes and applying a Dowker-type polygon inequality, then extend the approach to obtain a tight lower bound on the circumradius of the Minkowski sum of planar symmetric convex bodies and, more generally, a dimension-independent bound via intrinsic volumes. The main contributions include the precise bound , a sharp inequality for in terms of for symmetric planar bodies, and a best-possible absolute constant in the general d-dimensional setting, together with discussions of equality cases and potential generalizations to non-symmetric bodies and higher dimensions. The work connects discrete vector-sum problems with convex-geometric isoperimetric questions, yielding tight, widely applicable inequalities with implications for the geometry of zonotopes and Minkowski sums.

Abstract

We give a sharp lower bound to the largest possible Euclidean norm of signed sums of 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 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.

Paper Structure

This paper contains 4 sections, 4 theorems, 15 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Signed Sums of Planar Vectors
  4. Circumradius of Minkowski Sums of Convex Bodies

Key Result

Proposition 2.1

Let $P \subset \mathbb{R}^2$ be a convex $m$-gon for integers $1 \leq m \leq n$. Then with equality if and only if $P$ is a regular $n$-gon or a singleton.

Figures (1)

  • Figure 1: Two examples for Theorem \ref{['thm:planar_sym_sum_rad']}: $K^1$ (dashed, blue), $K^2$ (dotted, red), $K^1+K^2$ (dash-dotted, purple), $R(K^1) \mathbb{B}^2 = R(K^2) \mathbb{B}^2$ and $R(K^1+K^2) \mathbb{B}^2$ (solid, black). The constant in Theorem \ref{['thm:planar_sym_sum_rad']} for $n=2$ equals $\frac{1}{\sqrt{2}}$. The left-hand example shows that the necessary equality condition in Theorem \ref{['thm:planar_sym_sum_rad']} cannot be reduced to $K^1$ and $K^2$ consisting of only the described segments. The right-hand example shows that the necessary condition is, in general, not sufficient for equality.

Theorems & Definitions (9)

  • Proposition 2.1
  • proof
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • proof
  • Theorem 4.1
  • proof
  • Remark 4.2