Given an book on formulas (i.e., a partial map on **SL** to rational numbers of [0,1]), deciding if it is coherent is an **NP**-complete problem. The proof essentially uses Carathéodory theorem which characterizes the points of a convex set: given a convex set **C**=cl-co(**X**) whose affine dimension is *n*, then *x* belongs to C *if and only if* there is a finite subset **Y** of **X** of cardinality at most *n+1*, such that *x* is a convex combination of the elements of **Y**. Indeed a book is coherent if and only if it belongs to cl-co(**H**) being **H** the set of logical valuations.

Moving from coherence to strict-coherence essentially boils down, in geometrical terms, to providing a characterization, *á la* Carathéodory, for the *relative interior* of cl-co(**H**). Steinitz theorem gives a (unfortunately useless) direction: if a point *x* belongs to relint cl-co(**H**), then there exists a finite subset **K** of **H** of cardinality at most *2n* such that *x* belongs to relint cl-co(**K**). The converse direction (which is *key* for the decidability of a strictly coherent book) is trivially false: consider a cube **C** in R3 and pick a point *x* lying in the relint of a face of **C**. Then *x* belongs to the relative interior of a square, but not to the relint of the cube.

In the last weeks I’m just trying to extend (in certain sense) Steinitz theorem to obtain a characterization of the relative interior of a convex set. I don’t know if I will get it, but I like very much the drawings.