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Abstract: This is a joint work with I. Nasonov.
It is conjectured since long that for any convex body $P\subset \mathbb{R}^n$ there exists a point in its interior which belongs to at least $2n$ normals from different points on the boundary of $P$. The conjecture is known to be true for $n=2,3$ (E. Heil, 1985) and $n=4$ (J. Pardon, 2012).
We treat the same problem for convex polytopes and prove that each simple polytope in $\mathbb{R}^3$ has a point in its interior with 10 normals to the boundary. This is an exact bound: there exists a tetrahedron with at most 10 normals from a point in its interior. The proof is based on Morse–Cerf theory adjusted for polytopes.
Bio: Professor Gaiane Panina is an outstanding expert in Geometry and Topology.
Prof. Panina was born in Leningrad, now Saint Petersburg (she is a granddaughter of Acad. Victor Ambartsumian, one of the most known Soviet Astronomers). She graduated from Leningrad State University where she has got her PhD thesis in 1989.
In 2007, she has got her habilitation (Doctor of Science degree) with the thesis named 'Virtual polytopes'. Her research interests cover a broad area including but not are limited to: combinatorial geometry, polyhedral combinatorics, polygons, spaces of moduli, discrete Morse theory, universality, hinged mechanisms.
Now Prof. Panina works at Steklov Mathematical Institute of Russian Academy of Sciences, combining this with teaching at Saint Petersburg State University (Department of Mathematics and Computer Science) and PDMI Physics and Mathematics Club.
