Description of polygonal regions by polynomials of bounded degree

TL;DR

This paper demonstrates that convex polygons can be efficiently represented by polynomial inequalities with bounded degree, achieving near-optimal bounds on the number of inequalities needed, which advances understanding of polynomial descriptions of convex regions.

Contribution

It introduces a new method for representing convex polygons using polynomial inequalities with bounded degree, providing asymptotically optimal bounds on the number of inequalities required.

Findings

Representation of convex polygons with polynomial inequalities is near-optimal in the number of inequalities.

The method extends to interior representations with similar bounds.

Results hold for polynomials of degree up to a certain limit, depending on the polygon's edges.

Abstract

We show that every (possibly unbounded) convex polygon $P$ in $R^2$ with $m$ edges can be represented by inequalities $p_1 \ge 0,...,p_n \ge 0,$ where the $p_i$'s are products of at most $k$ affine functions each vanishing on an edge of $P$ and $n=n(m,k)$ satisfies $s(m,k) \le n(m,k) \le (1+\epsilon_m) s(m,k)$ with $s(m,k):=\max \{m/k,\log_2 m\}$ and $\epsilon_m \to 0$ as $m \to \infty$. This choice of $n$ is asymptotically best possible. An analogous result on representing the interior of $P$ in the form $p_1 > 0,..., p_n > 0$ is also given. For $k \le m/\log_2 m$ these statements remain valid for representations with arbitrary polynomials of degree not exceeding $k$.