A core defining at least partially a manufacturing form whose receiving surface, which is of essentially toroidal appearance, is designed to receive directly, by winding, profiled elements in the form of contiguous bands forming the inside surface of a tire, such core comprising a plurality of segments, each having transverse faces, said segments being circumferentially adjacent and arranged side by side, bearing against each other via their transverse faces, wherein a connecting radius between the transverse face and the receiving surface of a segment from among said plurality of segments 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.