Theorems on equipartition of a continuous mass distribution

Here are three samples of results.
Let $\mathbf m$ be a finite continuous mass distribution (an FCMD) in $\mathbb R^2$, and let $\ell=\{\ell_1,\dots,\ell_5\subset\mathbb R^2\}$ be 5 rays with common endpoint such that the sum of any two adjacent angles between them is at most $\pi$. Then $\mathbf m$ can be sibdivided into 5 parts at any prescribed ratio by an affine image of $\ell$.
For each FCMD $\mathbf m$ in $\mathbb R^n$ there exist $n$ mutually orthogonal hyperplanes any two of which subdivide $\mathbf m$ into 4 equal parts.
For any two FCMD's $\mathbf m_1$ and $\mathbf m_2$ in $\mathbb R^n$ with common center of symmetry $O$ there exist $n$ hyperplanes through $O$ any two of which subdivide both $\mathbf m_1$ and $\mathbf m_2$ into 4 equal parts.