Since the work of Zwart, it is known that cycling may occur in the cone splitting algorithm proposed by Tuy in 1964 to minimize a concave function over a polytope. In this paper, we show that despite this fact, Tuy's algorithm is convergent in the sense that it always finds an optimal solution. This is also true for a variant of Tuy's algorithm proposed by Gallo, in which a cone is split into a smaller subset of subcones (in term of inclusion). We show on an example that this variant may also cycle. The transformation of both algorithms into finite ones is discussed.