Fair Partitions

A substantial number of results and conjectures deal with the existence of a set of prescribed type which contains a fair share from each member of a finite collection of objects in a space, or the existence of partitions in which this is the case for every part. Examples include the Ham Sandwich Theorem in Measure Theory, the Hobby-Rice Theorem in Approximation Theory, the Necklace Theorem and Ryser’s Conjecture in Discrete Mathematics. The techniques in the study of these results combine combinatorial, topological, geometric, probabilistic and algebraic tools. This paper contains a brief description of the topic, focusing on several recent existence results and their algorithmic aspects. This is mainly a survey paper, but it also contains several novel results.