arXiv subjects
Publications and source records attributed to Illya Ivanov.
A convex $d$-dimensional polytope may be defined as a bounded intersection of closed halfspaces in $\mathbb{E}^d$ with interior. A polytope is primitive if omitting any halfspace renders the intersection unbounded. In this paper, we prove the Illumination Conjecture for the primitive polytopes: any primitive convex $d$-polytope $P$ can be illuminated with at most $2^d$ directions; even fewer if $P$ is not a linear image of a $d$-cube.
Given a convex domain $C$, a $C$-polygon is an intersection of $n\geq 2$ homothets of $C$. If the homothets are translates of $C$ then we call the intersection a translative $C$-polygon. This paper proves that if $C$ is a strictly convex domain with $m$ singular boundary points, then the number of singular boundary points a $C$-polygon has is between $n$ and $2(n-1)+m$. For a translative $C$-polygon we show the number of singular boundary points is between $n$ and $n+m$.