Extensions of theorems of Rattray and Makeev

TL;DR

This paper extends classical theorems by Rattray and Makeev to multiple maps and measures using orthonormal k-frames, and connects these results to embedding problems in topology.

Contribution

It generalizes existing theorems to multiple functions and measures with orthonormal k-frames, and links these extensions to topological embedding problems.

Findings

Theorems hold for several maps and measures simultaneously.

New results on partitioning measures with orthogonal hyperplanes.

Connections established between these theorems and projective space embeddings.

Abstract

We consider extensions of the Rattray theorem and two Makeev's theorems, showing that they hold for several maps, measures, or functions simultaneously, when we consider orthonormal $k$-frames in $\R^n$ instead of orthonormal basis (full frames). We also present new results on simultaneous partition of several measures into parts by $k$ mutually orthogonal hyperplanes. In the case $k=2$ we relate the Rattray and Makeev type results with the well known embedding problem for projective spaces.