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A subset of C2 having the properties in the title is constructed. If K is a compact subset of Cn, let ir(K) and p(K) denote the polynomially convex hull of K and the rationally convex hull of K, respectively. We then have ir(K) D p(K) D K. If K has zero two-dimensional Hausdorff measure, it is well known that p(K) = K. If F is a compact Jordan arc (homeomorph of [0, 1]) with finite one-dimensional Hausdorff measure, a deep result of Alexander [A] states that ir(F) = F. Thus Gamelin [G] has asked the following questions: (1) If F is a compact Jordan arc in Cn that has finite two-dimensional Hausdorff measure, does it follow that p(F) = F? (2) If a E (1, 2] and F is a compact Jordan arc in Cn that has finite a-dimensional Hausdorff measure, does it follow that ir(F) = F? The purpose of this note is to point out that question 1 can be answered by combining a recent theorem of Uy [U] with a well-known construction of Wermer [W] and Rudin [R]. This also gives an answer to question 2 in case a = 2. Theorem. If n > 2, there exists a compact Jordan arc F C Cn that has finite two-dimensional Hausdorff measure, but p(F) :A F. (In particular, r(F) :A F.) Proof. We may assume n = 2. Let E be a totally disconnected compact subset of C with positive two-dimensional Lebesgue measure. From the theorem of Uy [U] there exists a bounded Lipschitz function f: C C such that the restriction f I C\E is holomorphic and nonconstant, and f(oo) = 0. We define M: C -, C2 by M(z) (_ f(z), zf(z)), and note that M is injective on the complement of f-1 (0). Then the image X = M(E) is totally disconnected (in fact, if Y denotes the quotient space obtained from E by identifying all points of E n f-I (0), then Y is totally disconnected and M induces a homeomorphism from Y onto X.) It then follows from a theorem of Antoine [An] that there is a Jordan arc F C C2 containing X such that F\X is locally polygonal. This property, and the fact that M is Lipschitz on E, show that the set F = Received by the editors October 23, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 32E30; Secondary 32E20. Research was supported by NSERC of Canada and FCAR du Quebec. (D 1992 American Mathematical Society 0002-9939/92 $1.00 + $.25 per page

An arc of finite 2-measure that is not rationally convex