  In this paper we prove the existence of H\"{o}lder continuous terminal
embeddings of any desired $X \subseteq \mathbb{R}^d$ into $\mathbb{R}^{m}$ with
$m=\mathcal{O}(\varepsilon^{-2}\omega(S_X)^2)$, for arbitrarily small
distortion $\varepsilon$, where $\omega(S_X)$ denotes the Gaussian width of the
unit secants of $X$. More specifically, when $X$ is a finite set we provide
terminal embeddings that are locally $\frac{1}{2}$-H\"{o}lder almost
everywhere, and when $X$ is infinite with positive reach we give terminal
embeddings that are locally $\frac{1}{4}$-H\"{o}lder everywhere sufficiently
close to $X$ (i.e., within all tubes around $X$ of radius less than $X$'s
reach). When $X$ is a compact $d$-dimensional submanifold of $\mathbb{R}^N$, an
application of our main results provides terminal embeddings into
$\tilde{\mathcal{O}}(d)$-dimensional space that are locally H\"{o}lder
everywhere sufficiently close to the manifold.
