Structure of Null Sets in the Plane and Applications

Infinitesimal Structure of Differentiability Spaces, and Metric Differentiation

“…The previous discussion on the properties of the fragments inΓ Xn implies that the support spt Q R of Q R is a subset of S R ⊂ Ω∞ and that point (2) in the statement of this Theorem follows from (7.78). We now observe that:…”

“…Alberti representations were introduced in [1] to prove the so-called rank-one property for BV functions; they were later applied to study the di erentiability properties of Lipschitz functions f : R N → R [2,3] and have recently been used to obtain a description of measures in di erentiability spaces [10]. We rst give an informal de nition.…”

A Note on Alberti's Rank-One Theorem

“…Recently, Alberti, Csorniey and Preiss, (see [2]) have proposed a different proof of the Rank-one Theorem. This new proof uses as well the coarea formula, but it avoids Lemma 1.5, and relies instead on a general covering result for Lebesgue-null sets of the plane.…”

“…This new proof uses as well the coarea formula, but it avoids Lemma 1.5, and relies instead on a general covering result for Lebesgue-null sets of the plane. Let us mention, in passing, that this last result has many other deep implications in real analysis and geometric measure theory; see [2].…”

The Lip-lip equality is stable under blow-up

“…In connection with embeddings into RNP-Banach spaces, Cheeger and Kleiner [CK09] showed that if (X, µ) is a PI-space the fibres of T X are spanned by "tangent vectors" to Lipschitz curves. Putting T X and T * X on a complete equal footing has required substantial effort: Bate's beautiful work [Bat12,Bat15] on Alberti representations in differentiability spaces, which was partly motivated by a deep structure theory for measures and sets in R n developed by Alberti, Csörnyei and Preiss [ACP05,ACP10], and the formulation of metric differentiation for differentiability spaces [CKS15], which was partly motivated by unpublished results of Cheeger and Kleiner on metric differentiation in PI-spaces, and unpublished results of mine on prescribing the norms on T X and T * X.…”