In convex geometry, the Blaschke surface area measure on the boundary of a convex domain can be interpreted in terms of the complexity of approximating polyhedra. This approach is formulated in the holomorphic setting to establish an alternate interpretation of Fefferman's hypersurface measure on boundaries of strongly pseudoconvex domains in C 2 . In particular, it is shown that Fefferman's measure can be recovered from the Bergman kernel of the domain.1991 Mathematics Subject Classification. 32T15.
The totally-real embeddability of any 2k-dimensional compact manifold M into C n , n ≥ 3k, has several consequences: the genericity of polynomially convex embeddings of M into C n , the existence of n smooth generators for the Banach algebra C(M ), the existence of nonpolynomially convex embeddings with no analytic disks in their hulls, and the existence of special plurisubharmonic defining functions. We show that these results can be recovered even when n = 3k − 1, k > 1, despite the presence of complex tangencies, thus lowering the known bound for the optimal n in these (related but inequivalent) questions.pages 897-915, 1995.
Motivated by the theory of Hausdorff measures, we propose a new construction of the Fefferman hypersurface measure. This construction reveals the existence of non-trivial Fefferman-type measures on the boundary of some domains-such as products of balls-which are outside the purview of Fefferman's original definition. We also show that these measures enjoy certain transformation properties under biholomorphic mappings.
We establish a characterization for an m-manifold M to admit n functions f1,...,fn and n ′ functions g1, ..., g n ′ in C ∞ (M ) so that every element of C k (M ) can be approximated by rational combinations of f1, ..., fn and polynomial combinations of g1, ..., g n ′ . As an application, we show that the optimal value of n and n ′ for all manifolds of dimension m is ⌊ 3m 2 ⌋, when k ≥ 1 and m ≥ 2.
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