A core (N) defining at least partially a manufacturing form whose receiving surface (S), which is of essentially toroidal appearance, is designed to receive directly, by winding, profiles (B) in the form of contiguous bands forming the inside surface of a tire, comprising a plurality of segments (10, 10d, 10i), each having transverse faces (110d, 111d, 110i, 111i), said segments being circumferentially adjacent and located side by side, bearing against each other via their transverse faces, in which core the connecting radius (r) between the transverse face and the receiving surface (S) of a segment (10d, 10i) is between 0.8 mm and 1.5 mm, along all or part of the length of the edge formed by the intersection of these two surfaces.