exercise env
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2025-08-05 09:37:45 +08:00
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@@ -15,10 +15,10 @@
For errata and more stuff, see \url{https://sarielhp.org/book/}
\section{Grid}
\begin{oneshot}{1.1}
\begin{exercise}\label{ex1.1}
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}
\end{exercise}
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}
@@ -36,9 +36,9 @@ n &\geq 1/\vol(1b^d)\\
The last line is greater than $(\sqrt{d}/5)^d$ for large enough $d$.
\end{proof}
\begin{oneshot}{1.2}
\begin{exercise}
Compute clustering radius
\end{oneshot}
\end{exercise}
\end{document}