4 problems

Bezdek's illumination conjecture. There exists a positive constant $\varepsilon$ such that

Borisenko's conjecture. Among all such sets, the sets maximizing surface area are precisely the lenses of volume $V$.

Contraction dual-volume conjecture. Then

Bezdek's conjectures. The two volume extremality statements remain true when volume is replaced by $V_k$: at fixed $V_k$, the $(n-1)$-lens minimizes in-radius, and the $1$-lens max…