In this paper we consider the optimization of the shape of a cavity that intrudes into a solid conducting wall. This intrusion may be regarded as a "negative fin", i.e., the outside-in version of a conductive fin the shape of which is to be optimized. The objective is to minimize the global thermal resistance between the solid and the cavity. The cavity is rectangular, with fixed volume and variable aspect ratio. The cavity shape is optimized for two sets of thermal conditions for the solid wall: uniform heat generation, and uniform heat flux on the outer surfaces of the solid wall. The optimized cavity shape is practically independent of the thermal conditions. The cavity shape is optimal when it penetrates the conducting wall completely. In the second part of the paper we optimized a more complex intrusion: a cavity shaped as a T. The performance of the T-shaped cavity is superior to that of the finger-shaped cavity optimized in the first part of the paper. (C) 2004 Elsevier Ltd. All rights reserved.