# A convex cover for closed unit curves has area at least 0.0975

## TL;DR

This paper establishes a new lower bound of 0.0975 for the minimal area of convex sets covering all closed plane curves of unit length, improving previous bounds through geometric and numerical methods.

## Contribution

It introduces a combined geometric and numerical approach to improve the lower bound on the area of convex covers for unit curves.

## Key findings

- Lower bound of 0.0975 for convex covers of unit curves
- Improved previous lower bound of 0.096694
- Bounds for convex hulls of specific shapes like circle, triangle, and rectangle

## Abstract

We combine geometric methods with numerical box search algorithm to show that the minimal area of a convex set on the plane which can cover every closed plane curve of unit length is at least 0.0975. This improves the best previous lower bound of 0.096694. In fact, we show that the minimal area of convex hull of circle, equilateral triangle, and rectangle of perimeter $1$ is between 0.0975 and 0.09763.