On the Whitney Extension-Interpolation-Alignment problem for almost isometries with small distortion in $Bbb R^D$

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].

On Smooth Whitney Extensions of almost isometries with small distortion, Interpolation and Alignment in $Bbb R^D$-Part 1

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 $mathbb R^D$.

Shortest Path through Random Points

Let $(M,g_1)$ be a complete $d$-dimensional Riemannian manifold for $d > 1$. Let $mathcal X_n$ be a set of $n$ sample points in $M$ drawn randomly from a smooth Lebesgue density $f$ supported in $M$. Let $x,y$ be two points in $M$. We prove that the normalized length of the power-weighted shortest path between $x, y$ through $mathcal X_n$ converges almost surely to a constant multiple of the Riemannian distance between $x,y$ under the metric tensor $g_p = f^{2(1-p)/d} g_1$, where $p > 1$ is the power parameter.