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+\documentclass[12pt]{article}
+% \usepackage{chao}
+\usepackage[sans]{xenotes}
+% \usepackage{natbib}
+
+\title{Exercises in Sariel Har-Peled's \\ \textit{Geometric Approximation Algorithms}}
+\author{}
+\date{}
+
+\begin{document}
+\maketitle
+% \tableofcontents
+% \newpage
+
+For errata and more stuff, see \url{https://sarielhp.org/book/}
+
+\section{Grid}
+\begin{oneshot}{1.1}
+Let $P$ be a max cardinality point set contained in the $d$-dimensional unit hypercube such that the smallest distance of point pairs in $P$ is 1. Prove that 
+\[\left( \floor{\sqrt{d}}+1 \right)^d \leq |P|\leq \left( \ceil{\sqrt{d}}+1 \right)^d. \]
+\end{oneshot}
+hmm... the first exercise in this book is wrong. See \url{https://sarielhp.org/book/errata.pdf}.
+The stated lowerbound is actually an upperbound.
+\begin{proof}
+We evenly partition the $[0,1]$ interval into $m=\left( \floor{\sqrt{d}}+1 \right)$ small segments for each of the $d$ axes. The unit hypercube is partitioned into $m^d$ cells. The length of each cell's diagonal is $\sqrt{\frac{d}{m^2} }< 1$. Thus there is at most one point of $P$ in each cell and there are $\left( \floor{\sqrt{d}}+1 \right)^d$ cells.
+
+For lowerbound, one can construct a solution of size $2^d$ by selecting vertices of the hypercube. For sufficient large $d$ one can find a solution of size $(\sqrt{d}/5)^d$.\footnote{Exercise 1.1 (C) in \url{https://sarielhp.org/book/chapters/min_disk.pdf}}
+Let point set $P$ be the optimal solution and let $n=|P|$. We place a $d$-dimensional unit sphere around each point of $P$. These $n$ spheres must cover the unit hypercube since otherwise we can add more points into $P$. Thus one has $n\vol(1b^d)\geq 1$.
+\begin{equation*}
+\begin{aligned}
+n &\geq 1/\vol(1b^d)\\
+    &= \frac{\Gamma(d/2+1)}{\pi^{d/2}}\\
+    &\geq \sqrt{2\pi/(d/2+1)} (\frac{\sqrt{d}}{\sqrt{2e\pi}})^{d}
+\end{aligned}
+\end{equation*}
+The last line is greater than $(\sqrt{d}/5)^d$ for large enough $d$.
+\end{proof}
+
+\begin{oneshot}{1.2}
+Compute clustering radius
+\end{oneshot}
+
+
+\end{document}