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Register a new userOn the Whitney Extension-Interpolation-Alignment problem for almost isometries with small distortion in $Bbb R^D$
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Let $Dgeq 2$, $Ssubset mathbb R^D$ be finite and let $phi:Sto mathbb R^D$ with $phi$ a small distortion on $S$. We solve the Whitney extension-interpolation-alignment problem of how to understand when $phi$ can be extended to a function $Phi:mathbb R^Dto mathbb R^D$ which is a smooth small distortion on $mathbb R^D$. Our main results are in addition to Whitney extensions, results on interpolation and alignment of data in $mathbb R^D$ and complement those of [14,15,20].
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In this paper, we study the following problem: Let $Dgeq 2$ and let $Esubset mathbb R^D$ be finite satisfying certain conditions. Suppose that we are given a map $phi:Eto mathbb R^D$ with $phi$ a small distortion on $E$. How can one decide whether $phi$ extends to a smooth small distortion $Phi:mathbb R^Dto mathbb R^D$ which agrees with $phi$ on $E$. We also ask how to decide if in addition $Phi$ can be approximated well by certain rigid and non-rigid motions from $mathbb R^Dto mathbb R^D$. Since $E$ is a finite set, this question is basic to interpolation and alignment of data in $mathbb R^D$.
In this memoir, we develop a general framework which allows for a simultaneous study of labeled and unlabeled near alignment data problems in $mathbb R^D$ and the Whitney near isometry extension problem for discrete and non-discrete subsets of $mathbb R^D$ with certain geometries. In addition, we survey related work of ours on clustering, dimension reduction, manifold learning, vision as well as minimal energy partitions, discrepancy and min-max optimization. Numerous open problems in harmonic analysis, computer vision, manifold learning and signal processing connected to our work are given. A significant portion of the work in this memoir is based on joint research with Charles Fefferman in the papers [48], [49], [50], [51].
Let $ f $ be a real-valued function on a compact subset in $ mathbb{R}^n $. We show how to decide if $ f $ extends to a nonnegative and $ C^1 $ function on $ mathbb{R}^n $. There has been no known result for nonnegative $ C^m $ extension from a general compact set $ E $ when $ m > 0 $. The nonnegative extension problem for $ m geq 2 $ remains open.
High proved the following theorem. If the intersections of any two congruent copies of a plane convex body are centrally symmetric, then this body is a circle. In our paper we extend the theorem of High to spherical and hyperbolic planes. If in any of these planes, or in ${Bbb R}^2$, there is a pair of closed convex sets with interior points, and the intersections of any congruent copies of these sets are centrally symmetric, then, under some mild hypotheses, our sets are congruent circles, or, for ${Bbb R}^2$, two parallel strips. We prove the analogue of this statement, for $S^d$, ${Bbb R}^d$, $H^d$, if we suppose $C^2_+$: again, our sets are congruent balls. In $S^2$, ${Bbb R}^2$ and $H^2$ we investigate a variant of this question: supposing that the numbers of connected components of the boundaries of both sets are finite, we exactly describe all pairs of such closed convex sets, with interior points, whose any congruent copies have an intersection with axial symmetry (there are 1, 5 or 9 cases, respectively).
In this article we study the validity of the Whitney $C^1$ extension property for horizontal curves in sub-Riemannian manifolds endowed with 1-jets that satisfy a first-order Taylor expansion compatibility condition. We first consider the equiregular case, where we show that the extension property holds true whenever a suitable non-singularity property holds for the input-output maps on the Carnot groups obtained by nilpotent approximation. We then discuss the case of sub-Riemannian manifolds with singular points and we show that all step-2 manifolds satisfy the $C^1$ extension property. We conclude by showing that the $C^1$ extension property implies a Lusin-like approximation theorem for horizontal curves on sub-Riemannian manifolds.
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