We combine geometric methods with a numerical box search algorithm to show that the minimal area of a convex set in the plane which can cover every closed plane curve of unit length is at least [Formula: see text]. This improves the best previous lower bound of [Formula: see text]. In fact, we show that the minimal area of the convex hull of circle, equilateral triangle, and rectangle of perimeter [Formula: see text] is between [Formula: see text] and [Formula: see text].
We improve a lower bound for the smallest area of convex covers for closed unit curves from 0.0975 to 0.1, which makes it substantially closer to the current best upper bound 0.11023. We did this by considering the minimal area of convex hull of circle, line of length 1 2 , and rectangle with side 0.1727 × 0.3273. By using geometric methods and the Box search algorithm, we proved that this area is at least 0.1. We give informal numerical evidence that the obtained lower bound is close to the limit of current techniques, and substantially new idea is required to go significantly beyond 0.10044.
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