Theorems of Borsuk-Ulam type for flats and common transversals

TL;DR

This paper extends Borsuk-Ulam type theorems to the topology of k-flats in Euclidean space, deriving new results on common transversals and measure partitions by hyperplanes.

Contribution

It introduces novel topological theorems for flats in Euclidean space, linking them to transversals and measure partition problems.

Findings

Results on the topology of k-flats similar to Borsuk-Ulam theorem

Corollaries on common transversals for compact sets

Insights into measure partitions by hyperplanes

Abstract

In this paper some results on the topology of the space of $k$-flats in $\mathbb R^n$ are proved, similar to the Borsuk-Ulam theorem on coverings of sphere. Some corollaries on common transversals for families of compact sets in $\mathbb R^n$, and on measure partitions by hyperplanes, are deduced.