The purpose of this note is to bring into attention an apparently forgotten result of C. M. Petty: a convex body has minimal surface area among its affine transformations of the same volume if, and only if, its area measure is isotropic. We obtain sharp affine inequalities which demonstrate the fact that this “surface isotropic” position is a natural framework for the study of hyperplane projections of convex bodies.

Abstract. We study the maximal spaces of strong continuity on BM OA and the Bloch space B for semigroups of composition operators. Characterizations are given for the cases when these maximal spaces are V M OA or the little Bloch B 0 . These characterizations are in terms of the weak compactness of the resolvent function or in terms of a specially chosen symbol g of an integral operator T g . For the second characterization we prove and use an independent result, namely that the operators T g are weakly compact on the above mentioned spaces if and only if they are compact.

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We revisit the boundedness of Hankel and Toeplitz operators acting on the Hardy space H 1 and give a new proof of the old result stating that the Hankel operator Ha is bounded if and only if a has bounded logarithmic mean oscillation. We also establish a sufficient and necessary condition for Ha to be compact on H 1 . The Fredholm properties of Toeplitz operators on H 1 are studied for symbols in a Banach algebra similar to C + H ∞ under mild additional conditions caused by the differences in the boundedness of Toeplitz operators acting on H 1 and H 2 . Mathematics Subject Classification (2000). Primary 47B35; Secondary 30D50, 30D55, 47A53.

Using complex methods combined with Baire's Theorem, we show that onesided extendability, extendability, and real analyticity are rare phenomena on various spaces of functions in the topological sense. These considerations led us to introduce the p-continuous analytic capacity and variants of it, p ∈ {0, 1, 2, . . .} ∪ {∞}, for compact or closed sets in C. We use these capacities in order to characterize the removability of singularities of functions in the spaces A p .