Abstract:

A rectangular matrix is said to be totally positive (nonnegative) if all minors are positive (nonnegative). We will discuss Schönberg's variation-diminishing property, generalized isoperimetric inequalities for convex curves in even-dimensional euclidean spaces and Loewner-Whitney integrability. Then we will review the combinatorial works of Lindström and Gessel-Viennot and give a proof of the conjecture by Björner that all g-theorem matrices are totally nonnegative.