5 problems

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…

Let $\Omega$ be a domain with star-shaped hypersurface boundary, and let $\widetilde W_j(\Omega)$ denote the modified quermassintegrals and $h_{\alpha,j}$ the functions appearing i…

Quermassintegral inequalities conjecture. There hold

For a convex body $K\subset\mathbb{R}^n$, define … where $B_K$ is the Euclidean ball with the same volume as $K$. Lutwak's Alexandrov–Fenchel conjecture. For all $p\in[-n,0]$, … Th…

Let $K\subset\mathbb{R}^n$ be a convex body containing the origin in its interior. For $0\leq l\leq n-2$ and $\xi\in S^{n-1}$, let $W_l\big((K-t\xi)\cap\xi^\perp\big)$ denote the…