Regular polyhedral subdivisions and their tight spans

This is a survey talk discussing the following objects. A point configuration, $P$, in Euclidean space and a height function give rise to polyhedral subdivision of the convex hull of $P$. Subdivisions which arise in this way are called regular (or coherent). These polyhedral complexes admit a dual, which is again a polyhedral complex. Special cases include the tight spans of finite metric spaces (studied by Bandelt and Dress) as well as all tropical linear spaces. In this way this talk also serves as a gentle introduction to tropical geometry through polyhedral geometry.