Decreasing involutions and ℤ2-means
Abstract
An n-mean on a topological space X is a continuous symmetric map p: Xn→ X satisfying p(x,…,x)=x for every x∈ X. If X is a G-space, such an n-mean is equivariant if p(gx1,…,gxn)=g p(x1,…,xn) for every g∈ G and x1,…,xn∈ X. For finite G, we prove that, in the presence of an equivariant n-mean with n a multiple of lvert Grvert, the ANE, AE, and AR properties imply their equivariant counterparts. We next consider ℤ2-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 ℤ2-absolute retracts.
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BibTeX
@misc{jonardperez2026decreasing,
title = {{Decreasing involutions and $\mathbbZ_2$-means}},
author = {Natalia Jonard-Pérez and Ananda López-Poo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32994},
eprint = {2609.32994},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32994},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Decreasing involutions and ℤ_2-means
AU - Natalia Jonard-Pérez
AU - Ananda López-Poo
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.32994
UR - https://arxiv.org/abs/2609.32994
AB - An n-mean on a topological space X is a continuous symmetric map p: X^n→ X satisfying p(x,…,x)=x for every x∈ X. If X is a G-space, such an n-mean is equivariant if p(gx_1,…,gx_n)=g p(x_1,…,x_n) for every g∈ G and x_1,…,x_n∈ X. For finite G, we prove that, in the presence of an equivariant n-mean with n a multiple of lvert Grvert, the ANE, AE, and AR properties imply their equivariant counterparts. We next consider 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 ℤ_2-absolute retracts.
ER -