The hamburger theorem

We generalize the ham sandwich theorem to $d+1$ measures in $\mathbb{R}^d$ as follows. Let $\mu_1,\mu_2, \dots, \mu_{d+1}$ be absolutely continuous finite Borel measures on $\mathbb{R}^d$. Let $\omega_i=\mu_i(\mathbb{R}^d)$ for $i\in [d+1]$, $\omega=\min\{\omega_i; i\in [d+1]\}$ and assume that $\sum_{j=1}^{d+1} \omega_j=1$. Assume that $\omega_i \le 1/d$ for every $i\in[d+1]$. Then there exists a hyperplane $h$ such that each open halfspace $H$ defined by $h$ satisfies $\mu_i(H) \le (\sum_{j=1}^{d+1} \mu_j(H))/d$ for every $i \in [d+1]$ and $\sum_{j=1}^{d+1} \mu_j(H) \ge \min(1/2, 1-d\omega) \ge 1/(d+1)$. As a consequence we obtain that every $(d+1)$-colored set of $nd$ points in $\mathbb{R}^d$ such that no color is used for more than $n$ points can be partitioned into $n$ disjoint rainbow $(d-1)$-dimensional simplices.Comment: 11 pages, 2 figures; a new proof of Theorem 8, extended concluding remark

Cutting the same fraction of several measures

We study some measure partition problems: Cut the same positive fraction of $d+1$ measures in $\mathbb R^d$ with a hyperplane or find a convex subset of $\mathbb R^d$ on which $d+1$ given measures have the same prescribed value. For both problems positive answers are given under some additional assumptions.Comment: 7 pages 2 figure

The New Hampshire, Vol. 108, No. 27 (May 2, 2019)

An independent student produced newspaper from the University of New Hampshire

Red Honey

Red Honey is a collection of conceptual poetry based around the transfeminine body, the traumatized body, and the queer body. It follows transformations, rebirths, and deaths to their ends. Nothing is inevitable. This is a spell book for those who wish to live again

December 9, 1982

The Breeze is the student newspaper of James Madison University in Harrisonburg, Virginia