Decreasing involutions and $\mathbb{Z}_{2}$-means

Abstract

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.