In this paper we provide new geometric invariants of surjective isometries between unit spheres of Banach spaces. Let $X,Y$ be Banach spaces and let $T:S_X\to S_Y$ be a surjective isometry. The most relevant geometric invariants under surjective isometries such as T are known to be the starlike sets, the maximal faces of the unit ball, and the antipodal points (in the finite-dimensional case). Here, new geometric invariants are found, such as almost flat sets, flat sets, starlike compatible sets, and starlike generated sets. Also, in this work, it is proved that if F is a maximal face of the unit ball containing inner points, then $T(-F)=-T\left(F\right)$. We also show that if $[x,y]$ is a non-trivial segment contained in the unit sphere such that $T\left(\right[x,y\left]\right)$ is convex, then T is affine on $[x,y]$. As a consequence, T is affine on every segment that is a maximal face. On the other hand, we introduce a new geometric property called property P, which states that every face of the unit ball is the intersection of all maximal faces containing it. This property has turned out to be, in a implicit way, a very useful tool to show that many Banach spaces enjoy the Mazur-Ulam property. Following this line, in this manuscript it is proved that every reflexive or separable Banach space with dimension greater than or equal to 2 can be equivalently renormed to fail property P.

中文翻译：

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单位球体间射影等距的几何不变量

在本文中，我们提供了 Banach 空间的单位球面之间的满射等距的新几何不变量。让$X,是$ 成为 Banach 空间，让 $吨：{秒}_X\to {秒}_{是}$是一个满射等距。已知在诸如T 的满射等距下最相关的几何不变量是类星集、单位球的最大面和对映点（在有限维情况下）。在这里，发现了新的几何不变量，例如几乎平坦集、平坦集、星状兼容集和星状生成集。此外，在这项工作中，证明了如果F是包含内点的单位球的最大面，那么$吨(-F)=-吨\left(F\right)$. 我们还表明，如果$[X,是]$ 是包含在单位球体中的非平凡段，使得 $吨\left(\right[X,是\left]\right)$是凸的，那么T是仿射的$[X,是]$. 因此，T在每个最大面的段上都是仿射的。另一方面，我们引入了一个称为属性P的新几何属性，它指出单位球的每个面都是包含它的所有最大面的交集。事实证明，这个性质以一种隐含的方式，是一个非常有用的工具，可以表明许多 Banach 空间都享有 Mazur-Ulam 性质。沿着这条线，在这篇手稿中，证明了每个维数大于或等于 2 的自反或可分离 Banach 空间都可以等价地重新归一化以失败P。