4 problems

Let $F$ be a distance function, and suppose that each irrationally sloped line in $\operatorname{skel}(F)$ is insignificant. Conjecture. These lines may be removed from the star bo…

Let $F$ be a distance function whose skeleton, denoted by $\operatorname{skel}(F)$, consists of infinitely many rationally sloped lines. In this case, a fundamental region is not w…

Gardner–Giannopoulos convexity conjecture. The body $C_pK$ is convex whenever $-1<p\leq 1$; alternatively, $C_pK$ is convex for every $p>-1$ if $K$ is symmetric.

Generalized Busemann intersection inequality. More generally, for an arbitrary real number $m$ such that the integral on the right-hand side exists in the Lebesgue sense,