Journal of Mathematical Economics 3 (1976) 103-105. 6 North-Holland Publishing Company A REMARX ON A SMOOTHNESS PROPERTY OF CONVEX, COMPLETE PREORDERS* Andreu MAS-COLELL University of California, Berkeley, Cal$, U.S.A. Received August 1975 Let 2 = A x A be a complete preorder on A, a closed, convex subset of R”. Assume (i) 2 is closed (continuity), and (ii) if x k y, then CIX + (1 - a)~ 2 y for all 0 s ci 6 1 (convexity). Denote and define a (‘gradient’) mapping gz : A + B by g&) = {qE B: if y k x, then qy 2 qx}; Sk(x) is a convex set and it has a well-defined dimension. For every m s n, let C,,, = {XE A: dim g2(x) 2 m}, and denote by p,,, the m-dimensional Hausdorff measure in R” [for the definition and some facts about the Hausdorff measure, see Federer (1969, 2. lo)] ; p,, coincides with the usual Lebesgue measure. We prove Theorem. For every m 5 n, p,,_,+z(C,,J = 0; in particular, p”(C,) = 0, i.e., gx is a (possibly degenerate) ray a.e. (in the sense ofLebesgue measure). The theorem gives an analog for convex, complete preorders of the a.e. differentiability properties of concave functions; convex, complete preorders play in ‘ordinal’ maximization problems, i.e., those where the value of the maximization criterion is inessential and only the maximizers matter, the same *Research support from NSF Grant SOC73-0565OAOl and SOC72-05551 A02 is gratefully acknowledged.