[grads] [Sem-coll] !!! Seminar today !!! 4/22/2008

[Snejana Abarji]

Dear Colleagues

Hope you have enjoyed the seminar yesterday.

You are cordially invited to attend the Allied Math seminar. The seminar 
will be given today by Prof. Janos Pach from Courant Institute, NYU. The 
title of the seminar is "Points Surrounding the Origin." The seminar will 
be held in E1-106 at 4:40 pm.

Please, point your attention to the non-standard time of the seminar.

Please, find below an introductory note by Michael Pelsmajer, who is the 
host of Janos Pach at IIT, and the abstract of Janos Pach's talk. Please, 
contact to Michael regarding the schedule of his visitor.

This promises to be a very interesting event.

We have great speakers at the Applied Math Colloquia and Seminar this 
week.

You are very welcome to attend.

Best regards

Snejana

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Janos Pach

Janos Pach is a research professor at New York University Courant 
Institute of Mathematical Sciences and holds a distinguished professorship 
at City University of New York City College & The Graduate Center.  He is 
on the editorial board of Combinatorics, Computational Geometry: Theory 
and Applications, Geombinatorics, Graphs and Combinatorics, SIAM Journal 
of Discrete Mathematics, Applied Mathematics Research eXpress, 
International Journal of Computer Mathematics, and he is an 
editor-in-chief of Discrete and Computational Geometry.  He won the 
Grunwald Medal (Bolyai Mathematical Society, 1982), the Ford Award (MAA 
1990), the Renyi Award and Academy Award (Hungarian Academy of Sciences, 
1993 & 1998). He was the principal speaker for the NSF/CBMS Regional 
Research Conference on Geometric Graph Theory in 2002, and he has 
coauthored a textbook and a research monograph in combinatorial geometry. 
Lastly, he is an excellent speaker.

Best,
Michael

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Tuesday, April 22, 2008
Janos Pach (Courant Institute, NYU)
"Points Surrounding the Origin"
4:40 pm E1 106
For d > 2 and n > d+1, let P = { p1, . . . , pn } be a set of points
in Rd whose convex hull contains the origin O in its interior. We show
that if P ∪ O is in general position, then there exists a d-tuple Q =
{ pi1, . . . , pid } ⊂ P such that O is not contained in the convex
hull of Q ∪ {p} for any p ∈ P \ Q. Generalizations of this property
are also considered.

We also show that for disjoint, non-empty, finite point sets A1, . . .
, Ad+1 in Rd in general position with respect to the origin, if the
origin is contained in the convex hull of Ai ∪ Aj for all 1 ≤ i < j ≤
d+1, then there is a simplex S containing the origin such that |S ∩
Ai| = 1 for every 1 ≤ i ≤ d+1. This is a generalization of Bárány's
colored Carathéodory theorem, and dually, it gives a spherical version
of Lovász' colored Helly theorem.

Joint work with Andreas Holmsen and Helge Tverberg.

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