递减对合与$\mathbb{Z}_{2}$-均值
Decreasing involutions and $\mathbb{Z}_{2}$-means
- Universidad Nacional Autónoma de México（墨西哥国立自治大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究拓扑空间上等变均值的存在性，证明有限群条件下ANE、AE、AR性质的等变版本，并利用递减对合在模格上构造显式等变2-均值，应用于紧凸体和超强制凸函数空间。
AI中文摘要：
拓扑空间$X$上的$n$-均值是一个连续对称映射$p\colon X^n\to X$，满足对每个$x\in X$有$p(x,\ldots,x)=x$。若$X$是一个$G$-空间，则这样的$n$-均值是等变的，如果对每个$g\in G$和$x_1,\ldots,x_n\in X$有$p(gx_1,\ldots,gx_n)=g p(x_1,\ldots,x_n)$。对于有限群$G$，我们证明，在存在一个等变$n$-均值且$n$是$\lvert G\rvert$的倍数的情况下，$\mathrm{ANE}$、$\mathrm{AE}$和$\mathrm{AR}$性质蕴含其等变对应性质。接下来我们考虑由拓扑格上的递减对合诱导的$\mathbb Z_2$-作用。我们首先证明，在模格上，这样的对合的每个不动点都产生一个显式的等变$2$-均值。此外，我们引入一种用于逼近几何平均的巴比伦迭代的格论版本，并证明，在关于序及其与拓扑的相互作用的适当条件下，只要格具有与序相容的$2$-均值，该迭代就产生一个等变$2$-均值。最后，我们将这些结果应用于紧凸体空间和超强制凸函数空间，在其中证明等变$2$-均值的存在性，并表明这两个空间都是$\mathbb Z_2$-绝对收缩核。
英文摘要：
An $n$-mean on a topological space $X$ is a continuous symmetric map $p\colon X^n\to X$ satisfying $p(x,\ldots,x)=x$ for every $x\in X$. If $X$ is a $G$-space, such an $n$-mean is equivariant if $p(gx_1,\ldots,gx_n)=g p(x_1,\ldots,x_n)$ for every $g\in G$ and $x_1,\ldots,x_n\in X$. For finite $G$, we prove that, in the presence of an equivariant $n$-mean with $n$ a multiple of $\lvert G\rvert$, the $\mathrm{ANE}$, $\mathrm{AE}$, and $\mathrm{AR}$ properties imply their equivariant counterparts. We next consider $\mathbb Z_2$-actions induced by decreasing involutions on topological lattices. We first prove that every fixed point of such an involution on a modular lattice yields an explicit equivariant $2$-mean. Furthermore, we introduce a lattice-theoretic version of the Babylonian iteration used to approximate the geometric mean and prove that, under suitable conditions on the order and its interaction with the topology, this iteration produces an equivariant $2$-mean whenever the lattice admits a $2$-mean compatible with the order. Finally, we apply these results to spaces of compact convex bodies and supercoercive convex functions, where we prove the existence of equivariant $2$-means and show that both spaces are $\mathbb Z_2$-absolute retracts.
