  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 x
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.
