Measure-based classifier performance evaluation

Atesselation (sigma) is called strongly normal, if it is normal (topological discs with intersections that are either empty or connected) and for any subset of cells C1.....,C(K), C* of the tesselation holds: if the intersection (intersection k i=1) C(i), of all C(i), is nonempty and each C, has nonempty intersection with C*, then the intersection C* intersection intersection k i=1) C(i), of all C, with C* is nonempty. This concept was introduced for polygonal or polyhedral cells in a recent paper by Saha and Rosenfeld, where they proved that it is equivalent to the topological property that any cell together with any set of neighbouring cells forms a simply connected set. Answering a question from their paper, it is shown here that at least in the plane the cells need not be convex polygons, but can be arbitrary topological discs. Also the property is already implied if all collections of three cells have this property, giving a simpler characterization and a connection to Helly-type theorems.