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# Polyhedra, polytopes and beyond (Fall 2016 Fejes Toth Lecture)

Asia Ivić Weiss (York University)

Friday, November 18, 2016 -

2:00pm to 2:45pm

MS 319

The Fall 2016 Fejes Toth Lecture will be given by Professor Asia Ivic Weiss of York University, Toronto, Canada...

## Combinatorics and Discrete Geometry Seminar

Speaker: Károly Bezdek (University of Calgary)
Oct 14 2016 -
4:00pm to 5:00pm
MS 319

The talk discusses a recent counterexample to a conjecture of Goodman and Goodman on non-separable finite families of positive homothetic convex bodies made in 1945.

## Online First

Authors: Károly Bezdek (U of C) and Zsolt Lángi (BUT, Budapest)

Journal: Discrete and Computational Geometry

Abstract: A finite family B $\mathcal{}$of balls with respect to an arbitrary norm in ℝ^is called a non-separable family if there is no hyperplane disjoint from UB that strictly separates some elements of B from all the other elements of B. In this paper we prove that if B is a non-separable family of balls of radii r_1r_2r_(n2) with respect to an arbitrary norm in ℝ^d${\mathbb{}}^{}$ (d2), then Ucan be covered by a ball of radius ∑_{i=1}^{n}r_i. This was conjectured by Erdős for the Euclidean norm and was proved for that case by Goodman and Goodman (Am Math Mon 52:494–498, 1945). On the other hand, in the same paper Goodman and Goodman conjectured that their theorem extends to arbitrary non-separable finite families of positive homothetic convex bodies in ℝ^dd2. Besides giving a counterexample to their conjecture, we prove that conjecture under various additional conditions.

Welcome to the University of Calgary's Centre for Computational & Discrete Geometry (CCDG), an academic research centre housed within the Department of Mathematics & Statistics and supported by the Canada Research Chairs Program, Canada Foundation for Innovation, Natural Sciences and Engineering Research Council of Canada, Faculty of Science and the Department of Mathematics and Statistics.

## Volume Inequalities for Arrangements of Convex Bodies, CRC Press

Professor Károly Bezdek and his PhD student Muhammad Khan are currently co-authoring the title:

“Volume inequalities for arrangements of convex bodies”

for the prestigious Discrete Mathematics and Its Applications series of the CRC Press.

Volume is a fundamental concept of geometry and plays a pivotal role in all the problems and applications discussed here. The above-mentioned book not only develop the theoretical foundations of the field but also emphasize its applications in geographic information systems, medical imaging and materials science. In addition, it aims at promoting the use of volume inequalities and computer-based techniques to resolve the most important and long-standing open questions in discrete geometry.

More details about the contents of the book will soon appear here. For now here is a poster advertising the book.