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Extensions of the notions of polynomially and rationally convex hulls areintroduced. Using these notions, a generalization of a result of Duval andLevenberg on polynomially convex hulls containing no analytic discs ispresented. As a consequence it is shown that there exists a Cantor set $X$ in${\mathbb C}^3$ with a nontrivial polynomially convex hull that contains noanalytic discs. Using this Cantor set, it is shown that there exist arcs andcurves in ${\mathbb C}^4$ with nontrivial polynomially convex hulls thatcontain no analytic discs. This answers a question raised a few years ago byBercovici and can be regarded as a partial answer to a question raised byWermer over 60 years ago. More generally, it is shown that every uncountable,compact subspace of a Euclidean space can be embedded as a subspace $X$ of${\mathbb C}^N$, for some N, in such a way as to have a nontrivial polynomiallyconvex hull that contains no analytic discs. In the case when the topologicaldimension of the space is at most one, $X$ can be chosen so as to have thestronger property that $P(X)$ has a dense set of invertible elements.

Spaces with polynomial hulls that contain no analytic discs