3 problems

Let $M$ be a smooth, closed, connected, embedded, $k$-convex, star-shaped hypersurface in the sphere $\mathbb{S}^{n+1}$ enclosing a bounded domain $\Omega$. Let…

Let $0\leq k\leq n$ and $0\leq m\leq k$. Let $\Omega$ be a star-shaped and $k$-convex domain in $\mathbb H^{n+1}$ with smooth boundary $M$. Let…

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…