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A criterion for pure unrectifiability of sets (via universal vector bundle)

Silvano Delladio (2011)

Annales Polonici Mathematici

Let m,n be positive integers such that m < n and let G(n,m) be the Grassmann manifold of all m-dimensional subspaces of ℝⁿ. For V ∈ G(n,m) let π V denote the orthogonal projection from ℝⁿ onto V. The following characterization of purely unrectifiable sets holds. Let A be an m -measurable subset of ℝⁿ with m ( A ) < . Then A is purely m-unrectifiable if and only if there exists a null subset Z of the universal bundle ( V , v ) | V G ( n , m ) , v V such that, for all P ∈ A, one has m ( n - m ) ( V G ( n , m ) | ( V , π V ( P ) ) Z ) > 0 . One can replace “for all P ∈ A” by “for m -a.e. P ∈...

A Non-Probabilistic Proof of the Assouad Embedding Theorem with Bounds on the Dimension

Guy David, Marie Snipes (2013)

Analysis and Geometry in Metric Spaces

We give a non-probabilistic proof of a theorem of Naor and Neiman that asserts that if (E, d) is a doubling metric space, there is an integer N > 0, depending only on the metric doubling constant, such that for each exponent α ∈ (1/2; 1), one can find a bilipschitz mapping F = (E; dα ) ⃗ ℝ RN.

A quantitative version of the isoperimetric inequality : the anisotropic case

Luca Esposito, Nicola Fusco, Cristina Trombetti (2005)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We state and prove a stability result for the anisotropic version of the isoperimetric inequality. Namely if E is a set with small anisotropic isoperimetric deficit, then E is “close” to the Wulff shape set.

A rough curvature-dimension condition for metric measure spaces

Anca-Iuliana Bonciocat (2014)

Open Mathematics

We introduce and study a rough (approximate) curvature-dimension condition for metric measure spaces, applicable especially in the framework of discrete spaces and graphs. This condition extends the one introduced by Karl-Theodor Sturm, in his 2006 article On the geometry of metric measure spaces II, to a larger class of (possibly non-geodesic) metric measure spaces. The rough curvature-dimension condition is stable under an appropriate notion of convergence, and stable under discretizations as...

A Sard type theorem for Borel mappings

Piotr Hajłasz (1994)

Colloquium Mathematicae

We find a condition for a Borel mapping f : m n which implies that the Hausdorff dimension of f - 1 ( y ) is less than or equal to m-n for almost all y n .

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