Reconstructing polytopes from projections

Series: 
Geometry Topology Seminar
Monday, November 28, 2016 - 14:00
1 hour (actually 50 minutes)
Location: 
Skiles 006
,  
Kent State University
,  
Organizer: 
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.