A thin elastic plate, supported along the boundary, is loaded either by a single concentrated force at a given internal point, or by two equal and opposite concentrated forces at given internal diametrically opposite points. In each problem, the compliance, defined as the work done by the load, can be reduced on stiffening the plate by the addition to the boundary of an elastic strip whose cross-section is variable but whose total volume is prescribed. The cross-sectional area that minimises the compliance provides the optimal reinforcement for both problems. The respective compliances are compared with those obtained when the plate's boundary is reinforced by a strip of uniform cross-section and of the given volume. © Springer-Verlag Berlin Heidelberg 2006.