TY - JOUR
T1 - High-dimensional Menger-type curvatures. Part I
T2 - Geometric multipoles and multiscale inequalities
AU - Lerman, Gilad
AU - Whitehouse, J. Tyler
PY - 2011
Y1 - 2011
N2 - We define discrete and continuous Menger-type curvatures. The discrete curvature scales the volume of a (d + 1)-simplex in a real separable Hubert space H, whereas the continuous curvature integrates the square of the discrete one according to products of a given measure (or its restriction to balls). The essence of this paper is to establish an upper bound on the continuous Menger-type curvature of an Ahlfors regular measure μ on H in terms of the Jones-type flatness of μ(which adds up scaled errors of approximations of μ by d-planes at different scales and locations). As a consequence of this result we obtain that uniformly rectifiable measures satisfy a Carleson-type estimate in terms of the Menger-type curvature. Our strategy combines discrete and integral multiscale inequalities for the polar sine with the "geometric multipoles" construction, which is a multiway analog of the well-known method of fast multipoles.
AB - We define discrete and continuous Menger-type curvatures. The discrete curvature scales the volume of a (d + 1)-simplex in a real separable Hubert space H, whereas the continuous curvature integrates the square of the discrete one according to products of a given measure (or its restriction to balls). The essence of this paper is to establish an upper bound on the continuous Menger-type curvature of an Ahlfors regular measure μ on H in terms of the Jones-type flatness of μ(which adds up scaled errors of approximations of μ by d-planes at different scales and locations). As a consequence of this result we obtain that uniformly rectifiable measures satisfy a Carleson-type estimate in terms of the Menger-type curvature. Our strategy combines discrete and integral multiscale inequalities for the polar sine with the "geometric multipoles" construction, which is a multiway analog of the well-known method of fast multipoles.
KW - Ahlfors regular measure
KW - Least squares d-planes
KW - Menger curvature
KW - Menger-type curvature
KW - Multiscale geometry
KW - Polar sine
KW - Recovering lowdimensional structures in high dimensions
KW - Uniform rectifiability
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U2 - 10.4171/RMI/645
DO - 10.4171/RMI/645
M3 - Article
AN - SCOPUS:84856025075
SN - 0213-2230
VL - 27
SP - 493
EP - 555
JO - Revista Matematica Iberoamericana
JF - Revista Matematica Iberoamericana
IS - 2
ER -