From repeated distances to the complexity of the union of convex bodies
Speaker:
Janos Pach, Renyi Institute of Mathematics
Date and Time:
Tuesday, July 11, 2023 - 11:00am to 11:30am
Location:
Fields Institute, Room 230
Abstract:
In 1946, Erdős raised an innocent looking question in the American Mathematical Monthly: At most how many times can the unit distance occur among n points in Euclidean d-space? 15 years later, he found an asymptotically tight answer in 4 and higher dimensions, but the problem is still open in the plane and in 3-space.
The following problem originates in robotics: At most how many cavities can be surrounded by the union of n convex bodies of a certain type? It turns out that the above questions are intimately connected to each other. After a whirlwind tour of the subject, we describe some recent results and list a few open problems.