In this note various geometric properties of a Banach space $X$ arecharacterized by means of weaker corresponding geometric properties involvingan ultrapower $X^\mathcal{U}$. The characterizations do not depend on theparticular 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 canonicalinclusion $j\colon X \to X^\mathcal{U}$) in $B_{X^\mathcal{U}}$ is an extremepoint; a point $x\in S_X$ is LUR if and only if $j(x)$ is not contained in anynon-degenerate line segment of $S_{X^\mathcal{U}}$; a Banach space $X$ is UREDif and only if there are no $x,y \in S_{X^\mathcal{U}}$, $x\neq y$, with $x-y\in j(X)$.
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机译：在本说明中，Banach空间$ X $的各种几何特性是通过涉及超能力$ X ^ \ mathcal {U} $的较弱的对应几何特性来表征的。表征不取决于自由超滤器$ \ mathcal {U} $的特定选择。例如，当且仅当$ B_ {中的$ j（x）$（由canonicalinclusion $ j \冒号X \ to X ^ \ mathcal {U} $给出）时，S_X $中的点$ x \是MLUR点。 X ^ \ mathcal {U}} $是一个极端；当且仅当$ j（x）$不包含在$ S_ {X ^ \ mathcal {U}} $的任何非退化线段中时，S_X $中的点$ x \为LUR； Banach空间$ X $是UREDif，并且仅当S_ {X ^ \ mathcal {U}} $，$ x \ neq y $中没有$ x，y \，而j（X）$中有$ x-y \时才是UREDif。
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