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Geometric measure theory in Banach spaces. (English) Zbl 1064.46068
Johnson, W. B. (ed.) et al., Handbook of the geometry of Banach spaces. Volume 2. Amsterdam: North-Holland (ISBN 0-444-51305-1/hbk). 1519-1546 (2003).
The chapter under review is a remarkable account of the recent progress on geometric measure theory, illustrated and motivated by applications to Lipschitz maps and differentiability problems, in (mainly infinite-dimensional and separable) Banach spaces, written by a leader in this field. Technicalities are not fully displayed, so that reading the paper is pleasant and relatively easy. However, difficulties are clearly pointed out, both in existing proofs and in plausible approaches to open problems, and the reader is properly made aware of the technical and intricate nature of this branch of geometric functional analysis.
The topics which are discussed in this article include the following: 1. Rectifiability, density and currents, with a nice presentation of the not so well-known concept of current, with its recent introduction in infinite-dimensional ambient spaces. 2. Differentiability of measures, directional derivatives, differentiation in the sense of Skorokhod, of Tonelli, of Fomin, properties of Gaussian measures. 3. Characterization of measures by their values on balls in metric spaces or in Banach spaces, with or without conditions on the radii. 4. Various notions of negligible sets: Haar null sets, Aronszajn null sets, \(\Gamma\)-null sets. These notions aim at providing results of differentiability of Lipschitz functions outside a negligible set in an infinite-dimensional context. They appear, out of necessity, since there is no nonzero translation invariant measure on an infinite-dimensional Banach space: a very simple but fundamental fact. 5. Differentiability of convex functions, \(\sigma\)-porous sets, simultaneous Fréchet differentiability of a convex function and Gâteaux differentiability of a Lipschitz map from a separable Asplund space to a space with the Radon-Nikodym property, metric derivatives.
The reader is alerted on the deep differences between the finite-dimensional and the infinite-dimensional case, but made aware on the importance of finite-dimensional considerations in the proofs of many infinite-dimensional statements. Fascinating open problems are outlined or recalled, such as the following question, from the introduction: does every Lipschitz function from a separable Hilbert space to an \(n\)-dimensional space (\(n>1\)) have points of Fréchet differentiability? Solving positively this question would be an important step towards the solution of the great differentiability conjectures, which might imply, for instance, that two separable reflexive Lipschitz-isomorphic Banach spaces are linearly isomorphic – another open problem. This article is a pleasant invitation to get involved with these very challenging problems, and a reference that everyone who considers working in this area should read and meditate on.
For the entire collection see [Zbl 1013.46001].

46T12 Measure (Gaussian, cylindrical, etc.) and integrals (Feynman, path, Fresnel, etc.) on manifolds
46T20 Continuous and differentiable maps in nonlinear functional analysis
28A75 Length, area, volume, other geometric measure theory
28A78 Hausdorff and packing measures
28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
46G12 Measures and integration on abstract linear spaces
46-00 General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to functional analysis