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

• Autorzy: Silvano Delladio
• Języki publikacji: EN

Abstrakty

EN

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 $𝓗 ^m$-measurable subset of ℝⁿ with $𝓗 ^m(A) < ∞$. 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 $𝓗 ^{m(n-m)}({V ∈ G(n,m)|(V,π_V(P)) ∈ Z}) > 0$. One can replace "for all P ∈ A" by "for $𝓗 ^m$-a.e. P ∈ A".

Kategorie tematyczne

• 53A05: Surfaces in Euclidean space
• 28A75: Length, area, volume, other geometric measure theory
• 28A78: Hausdorff and packing measures
• 49Q15: Geometric measure and integration theory, integral and normal currents

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 2011
• Tom: 102
• Numer: 1
• Strony: 73-78

Daty

wydano

2011

Twórcy

autor

Silvano Delladio

• Dipartimento di Matematica, Università di Trento, Trento, Italy

Identyfikatory

DOI

10.4064/ap102-1-6