A subset \(E\) of a metric space \(X\) is called rectifiable if it can be covered up to a set of one-dimensional Hausdorff measure zero by countably many Lipschitz curves. On the other hand \(E\) is called purely unrectifiable if it intersects every Lipschitz curve in \(X\) in a set of one-dimensional Hausdorff measure zero. Amoung sets of finite one-dimensional Hausdorff measure in Euclidean space \(\mathbb R^{n}\) (\(n\geq 1\)), the Besicovitsch-Federer Projection Theorem characterises purely unrectifiable sets as those sets whose orthogonal projection onto almost every one-dimensional subspace of \(\mathbb R^{n}\) is of Lebesgue measure zero; see [P. Mattila, Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability. Cambridge: Univ. Press (1995; Zbl 0819.28004), Chapter 16].
The reviewed paper completely resolves the question of whether the Besicovitsch-Federer Projection Theorem extends to infinite dimensional Banach spaces in the negative. The natural analogue of the projection theorem in the setting of an infinite dimensional Banach space \(X\) proposes that a subset \(E\) of \(X\) with \(\mathcal{H}^{1}(E)<\infty\) is purely unrectifiable if and only if the set of those functionals \(\varphi\in X^{*}\) for which the set \(\varphi(E)\) has positive Lebesgue measure is small in some sense. T. De Pauw [Publ. Mat., Barc. 61, No. 1, 153–173 (2017; Zbl 1359.28004)] shows this statement to be false in \(\ell_{2}\), when the notion of smallness in \(X^{*}\) is determined by the class of Aronszajn-null sets. However, [loc. cit.] leaves open the question of whether the projection theorem is valid in other Banach spaces and with other notions of smallness, for example Haar null. In the paper under review, these questions are completely settled: The authors show that the projection theorem dramatically fails in every infinite dimensional, separable Banach space \(X\) with respect to any notion of smallness in \(X^{*}\). More precisely, they prove that every infinite dimensional, separable Banach space \(X\) contains a purely unrectifiable set \(E\) with \(\mathcal{H}^{1}(E)<\infty\) for which the set \(\varphi(E)\) has positive Lebesgue measure for every functional \(\varphi\in X^{*}\setminus \left\{0\right\}\). Note that this result cannot extend to non-separable Banach spaces because any set of finite \(\mathcal{H}^{1}\)-measure is separable.
A key ingredient in the proof is the construction of a sequence \((x_{n})_{n=1}^{\infty}\) in \(X\) which behaves like an unconditional basis in the sense that the series \(\sum_{n=1}^{\infty}c_{n}x_{n}\) converges for any sequence of real numbers \((c_{n})_{n=1}^{\infty}\) with \(\sup_{n\in\mathbb N}\left|c_{n}\right|\leq 1\). The construction of this sequence \((x_{n})_{n=1}^{\infty}\) is based on a construction of A. Dvoretzky and C. A. Rogers [Proc. Natl. Acad. Sci. USA 36, 192–197 (1950; Zbl 0036.36303)]. The authors then construct sets \(F=F(f)\) in \(X\) taking the form of the image of the interval \([0,1]\) under mappings \(f:[0,1]\to X\) of the form \[ f=\sum_{n=1}^{\infty}f_{n}\cdot x_{n}, \] where each \(f_{n}\) is a function \([0,1]\to[0,1]\). Such sets \(F=f([0,1])\) are shown to have finite \(\mathcal{H}^{1}\)-measure, be purely unrectifiable and admit projections \(\varphi(E)\) of positive Lebesgue measure for every functional \(\varphi\in \overline{\text{span}}\left\{x_{1},x_{2},\ldots \right\}^{*}\setminus \left\{0\right\}\). To prove pure unrectifiability of the sets \(F\) the authors make important use of a theorem of B. Kirchheim [Proc. Am. Math. Soc. 121, No. 1, 113–123 (1994; Zbl 0806.28004)]. By taking an appropriate countable union of rescaled versions of \(F\), the authors acquire a purely unrectifiable set \(E\) with \(\mathcal{H}^{1}(E)<\infty\) for which \(\varphi(E)\) has positive Lebesgue measure whenever \(\varphi\in X^{*}\setminus \left\{0\right\}\).
To conclude the paper, the authors investigate projections of rectifiable subsets of a general Banach space and extend a known property of rectifiable subsets of Euclidean spaces to the general Banach space setting. They prove that for any Banach space \(X\) and any rectifiable set \(E\subseteq X\) with \(\mathcal{H}^{1}(E)>0\) the set \(\varphi(E)\subseteq \mathbb R\) is of positive Lebesgue measure for every functional \(\varphi\in X^{*}\setminus \left\{0\right\}\) except possibly those \(\varphi\) belonging to a closed, linear and proper subspace \(Y^{*}\) of \(X^{*}\). This theorem is sharp in every Banach space, as can be seen by taking \(E\) equal to a one-dimensional, linear subspace of \(X\).