Strongly concave functions on root systems

Let F be a local complete field with discrete valuation, and let G be the group of F-points of a quasisplit group over F. Let ℬ be the Bruhat-Tits building of G and let A be an apartment of ℬ; we establish a link between bounded closed subsets of A and a special kind of functions on the relative root system Φ of G, which will be called strongly concave functions. This paper is devoted to the study of such functions.