Tverberg's theorem and shadows of simplices and convex bodies

Rade Živaljević

Helge Tverberg published more than forty years ago his original proof of the theorem which has been widely acclaimed and today bears his name. This beautiful result has been one of the most celebrated results of discrete geometry and, together with its relatives, still remains a central and one of the most intriguing results of geometric combinatorics. Here we give a reasonably non-technical presentation of this result having in mind a larger mathematical audience, particularly school teachers and their talented students, hoping that it may raise their interest for this very attractive area of mathematics. In the remaining part of the article we briefly visit some of other branches of convex geometry and outline how ``smashing'' and `slicing'' of convex bodies offers a deep insight into their structure and behavior.