The present invention relates to an improvement in the clearances 12 between a casing and the moving blades 11 of a rotor 1 of a turbomachine, in particular a casing provided for this purpose, and also to a turbomachine fitted with such a casing. The invention applies in particular to aircraft turbojet engines.It relates more particularly to a turbomachine inner casing 21 comprising a plurality of annular shrouds 23 that are provided with flanges 24 for joining two adjacent shrouds 23 together. According to the invention, this inner casing 21 further includes: at least one circumferential heat shield 5 covering the flanges 24 in contact with two adjacent shrouds 23; and a cavity 6 filled with a first thermal insulation 8, said cavity being located between a concave internal surface 51a, 52a of the heat shield 5 and the flanges 24. The heat shield 5 is formed by a plurality of sectors 50 joined end to end.