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

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.

41 pages, 15 figures, paper