We present a new, general constraint which, in principle, determines the superconformal UMR symmetry of 4d N = 1 SCFFs, and also 3d N = 2 SCFTs. Among all possibilities, the superconformal U (1) R is that which minimizes the coefficient, tau(RR), of its two-point function. Equivalently, the superconformal UMR is the unique one with vanishing two-point function with every non-R flavor symmetry. For 4d N = 1 SCFTs, tau(RR) minimization gives an alternative to a-maximization. tau(RR) minimization also applies in 3d, where no condition for determining the superconformal UMR had been previously known. Unfortunately, this constraint seems impractical to implement for interacting field theories. But it can be readily implemented in the Ads geometry for SCFTs with Ads duals. (c) 2005 Published by Elsevier B.V.