Uniform-to-proper duality of geometric properties of Banach spaces and their ultrapowers

In this note various geometric properties of a Banach space $X$ are characterized by means of weaker corresponding geometric properties involving an ultrapower $X^{\mathcal{U}}$. The characterizations do not depend on the particular choice of the free ultrafilter $\mathcal{U}$. For example, a point $x\in S_X$ is an MLUR point if and only if $j(x)$ (given by the canonical inclusion $j\text{\hspace{0.17em}⁣}:X\to X^{\mathcal{U}}$) in $B_{X^{\mathcal{U}}}$ is an extreme point; a point $x\in S_X$ is LUR if and only if $j(x)$ is not contained in any non-degenerate line segment of $S_{X^{\mathcal{U}}}$; a Banach space $X$ is URED if and only if there are no $x,y\in S_{X^{\mathcal{U}}}$, $x\mathrm{\ne }y$, with $x-y \in j(X)$.