AI 中文总结
研究希尔伯特空间中基尔施布劳恩扩张保持一致距离的条件，证明相关等价条件，给出有限维目标应用及无限维或特定维度关系下的一些结论，如数据处理特征、提升定理和不等式等。
AI 中文摘要
设\(X\)是实希尔伯特空间的一个子集，\(v\colon X\to Y\)，其中\(Y\)是实希尔伯特空间。我们证明以下条件等价：对于任意\(A\subset X\)，\(\rho\geq0\)，以及\(1 -\)利普希茨的\(u\colon A\to Y\)且对\(x\in A\)有\(\left\lVert u(x)-v(x)\right\lVert\leq\rho\)，存在\(1 -\)利普希茨扩张\(\widetilde u\colon X\to Y\)且对\(x\in X\)有\(\left\lVert \widetilde u(x)-v(x)\right \lVert\leq\rho\)；以及对每个\(1\leq k\leq\dim Y\)，当\(x_0,\ldots,x_k\in X\)，\(t_1,\ldots,t_k\geq0\)且\(\sum_{i=1}^k t_i = 1\)时，\(\left\lVert v(x_0)-\sum_{i=1}^k t_i v(x_i)\right\lVert \leq \left\lVert x_0-\sum_{i=1}^k t_i x_i \right \lVert\)。之前的必要性结果要求\(\dim Y\leq3\)或\(X\)的凸性。对于有限维目标，一个应用给出了连接瓦瑟斯坦和重心弱运输的有限分支层次结构的精确数据处理特征。如果\(Y\)是无限维或\(\dim\operatorname{Aff}X + 1\leq\dim Y\)，我们还得到了凸利普希茨函数的提升定理和转移凸庞加莱不等式且不增加常数。
英文摘要
Let $X$ be a subset of a real Hilbert space and let $v\colon X\to Y$, where $Y$ is a real Hilbert space. We prove that the following conditions are equivalent: whenever $A\subset X$, $ρ\geq0$, and $u\colon A\to Y$ is $1$-Lipschitz with $\left\lVert u(x)-v(x)\right\lVert\leqρ$ for $x\in A$, there is a $1$-Lipschitz extension $\widetilde u\colon X\to Y$ with $\left\lVert \widetilde u(x)-v(x)\right \lVert\leqρ$ for $x\in X$; and for every $1\leq k\leq\dim Y$, $$ \left\lVert v(x_0)-\sum_{i=1}^k t_i v(x_i)\right\lVert \leq \left\lVert x_0-\sum_{i=1}^k t_i x_i \right \lVert$$ whenever $x_0,\ldots,x_k\in X$, $t_1,\ldots,t_k\geq0$, and $\sum_{i=1}^k t_i=1$. Previous necessity results required $\dim Y\leq3$ or convexity of $X$. For finite-dimensional targets, an application gives an exact data processing characterisation for a finite branching hierarchy connecting Wasserstein and barycentric weak transport. If $Y$ is infinite-dimensional or $\dim\operatorname{Aff}X+1\leq\dim Y$, we also obtain a lifting theorem for convex Lipschitz functions and transfer convex Poincaré inequalities without increasing the constant.
Comments17 pages; comments are welcome
