Reconstructing polytopes from projections

Geometry Topology Seminar
Monday, November 28, 2016 - 2:00pm
1 hour (actually 50 minutes)
Skiles 006
Kent State University
We are going to discuss one of the open problems of geometric tomography about projections. Along with partial previous results, the proof of the problem below will be investigated.Let $2\le k\le d-1$ and let $P$ and $Q$ be two convex polytopes in ${\mathbb E^d}$. Assume that their projections, $P|H$, $Q|H$, onto every $k$-dimensional subspace $H$,  are congruent. We will show  that  $P$ and $Q$ or $P$ and $-Q$ are translates of each other. If the time permits, we also will discuss an analogous result for sections by showing that $P=Q$ or $P=-Q$, provided the  polytopes  contain the  origin in their interior and their sections, $P \cap H$, $Q \cap H$, by every $k$-dimensional subspace $H$, are congruent.