The ACO Seminar (2021–2022)

The relationship between combinatorial codes and their geometric realizations is partially captured by “embedding dimension” invariants. These are the minimum dimensions of Euclidean space in which one can find a realization of the code by sets with specified properties (e.g. closed convex sets). We describe relationships between three embedding dimensions: open convex, closed convex, and non-degenerate. Specifically, we characterize all triples of embedding dimensions that can arise from a given code. Along the way we explain two convex geometry tools which may be of more general interest: a “sunflower theorem” that we proved in 2018, and a more recent notion of “rigid structure” defined by Chan, Johnston, Lent, Ruys de Perez, and Shiu.