A rotor blade tip clearance control apparatus for a gas turbine engine includes a plurality of circumferentially distributed segments which form an annular shroud surrounding the outer tips of a row of rotor blades. A mechanical arrangement is operatively connected to the segments. Actuation of the arrangement causes the segments to move in a radial direction thereby controlling a clearance between the segments and the outer tips. A case cooling system supplies cooling air to an engine case to which the segments are mounted. The cooling air regulates thermal expansion of the case and thereby also controls the clearance between the segments and the outer tips.