### Extensions of theorems of Rattray and Makeev

#### 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 $\mathbb{R}^n$ instead of orthonormal bases (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.

when we consider orthonormal $k$-frames in $\mathbb{R}^n$ instead of orthonormal bases (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.

#### Keywords

Rattray's theorem; measure partition; Borsuk-Ulam type theorems

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