September 23, 3:30pm,

R. Amzi Jeffs, Carnegie Mellon University

Open, Closed, and Non-Degenerate Embedding Dimensions of Convex Codes

R. Amzi Jeffs, Carnegie Mellon University

Open, Closed, and Non-Degenerate Embedding Dimensions of Convex Codes

Abstract:

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.