Lower Bound for Convex Hull Area and Universal Cover Problems

TL;DR

This paper establishes new lower bounds for the area of convex hulls and universal covers for curves, advancing understanding of geometric covering problems with specific quantitative limits.

Contribution

It introduces novel lower bounds for convex hull areas and applies them to universal cover problems, providing concrete minimal area estimates.

Findings

Convex universal cover for a unit length curve has area ≥ 0.232239

Convex universal cover for a unit closed curve has area ≥ 0.0879873

New geometric bounds improve understanding of covering problems

Abstract

In this paper, we provide a lower bound for an area of the convex hull of points and a rectangle in a plane. We then apply this estimate to establish a lower bound for a universal cover problem. We showed that a convex universal cover for a unit length curve has area at least 0.232239. In addition, we show that a convex universal cover for a unit closed curve has area at least 0.0879873.