4 problems

Hausdorff-measure lower-bound conjecture. There exists a constant $C>0$, depending on $n$, $k$, $hR$, and an upper bound for $\mathscr{H}^k(E)$, such that

Reuleaux-triangle minimizer conjecture. The Reuleaux triangle is a minimizer for the fixed-diameter problem.

Fixed-diameter existence conjecture. The minimization problem

Fixed-perimeter nonattainment conjecture. The minimization problem