Asymptotically optimally doubling measures and Reifenberg flat sets with vanishing constant. (English) Zbl 1031.28004
The authors show that a subset of \(\mathbb{R}^n\) is well approximated by \(n\)-dimensional affine spaces (in the sense of Hausdorff distance) if and only if it supports a measure whose asymptotic doubling properties coincide with those of Lebesgue measure on \(\mathbb{R}^n\).
Reviewer: Sophia L.Kalpazidou (Thessaloniki)
MSC:
| 28A75 | Length, area, volume, other geometric measure theory |
| 49Q15 | Geometric measure and integration theory, integral and normal currents in optimization |
| 58C35 | Integration on manifolds; measures on manifolds |
References:
| [1] | Cheeger, J Differential Geom 46 pp 406– (1997) · Zbl 0902.53034 · doi:10.4310/jdg/1214459974 |
| [2] | David, Math Ann 315 pp 641– (1999) · Zbl 0944.53004 · doi:10.1007/s002080050332 |
| [3] | ; Uniformly distributed measures in Euclidean spaces. Preprint. · Zbl 1024.28004 |
| [4] | Kowalski, J Reine Angew Math 379 pp 115– (1987) |
| [5] | Kenig, Duke Math J 87 pp 509– (1997) · Zbl 0878.31002 · doi:10.1215/S0012-7094-97-08717-2 |
| [6] | Kenig, Ann of Math 150 pp 369– (1999) · Zbl 0946.31001 · doi:10.2307/121086 |
| [7] | Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability. Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge, 1995. · Zbl 0819.28004 |
| [8] | Multiple integrals in the calculus of variations. Die Grundlehren der mathematischen Wissenschaften, 130. Springer, New York, 1966. |
| [9] | Preiss, Ann of Math 125 pp 537– (1987) · Zbl 0627.28008 · doi:10.2307/1971410 |
| [10] | Reifenberg, Acta Math 104 pp 1– (1960) · Zbl 0099.08503 · doi:10.1007/BF02547186 |
| [11] | Toro, Notices Amer Math Soc 44 pp 1087– (1997) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
