6 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 $n\geq 2$, let $\Sigma\subset -{\mathbb R}^{n+1}_{+}$ be a convex hypersurface with capillary boundary supported on $\partial-{\mathbb R}_{+}^{n+1}$ at a contact angle…

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…

Let $rho= obreakfrac{1+|x|^2}{1-|x|^2}$ in the Poincaré model of $mathbb{H}^n$, and let $Sigma$ be a horospherically convex hypersurface in $mathbb{H}^n$. Write $Hk$ for its normal…

Let $\Sigma$ be a horospherically convex hypersurface in hyperbolic space $\mathbb H^n$, let $V$ be the hyperbolic weight, and let $\sigma_k$ denote the normalized $k$th elementary…