For a Banach space X, let $\mathcal{C}(X)$ be the cone consisting of all nonempty bounded closed convex subsets of X endowed with the Hausdorff metric. In this paper, we show that if one of the two Banach spaces X and Y is an Asplund space, then for every surjective isometry $f:\mathcal{C}(X)\to \mathcal{C}(Y)$, the restriction $f{|}_X$ is a surjective affine isometry from X to Y. If, in addition, one of X and Y is Fréchet smooth, or, locally uniformly convex, then $f(C)=\bigcup \{f(x):x\in C\}$ for all $C\in \mathcal{C}(X)$.

中文翻译：

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Asplund空间上Mazur-Ulam定理的凸集值版本

对于Banach空间X，让$\mathcal{C}（X）$是由赋予Hausdorff度量的X的所有非空有界封闭凸子集组成的圆锥。在本文中，我们表明，如果两个Banach空间X和Y之一是Asplund空间，则对于每个射影等距$F：\mathcal{C}（X）\to \mathcal{C}（ÿ）$，限制 $F{|}_X$是从X到Y的射影仿射等距图。另外，如果X和Y之一是Fréchet光滑的，或者局部均匀地凸，则$F（C）=\bigcup \{F（X）：X\in C\}$ 对全部 $C\in \mathcal{C}（X）$。