How to correctly shrink the bullets of itemize

Question

How can I shrink the bullets of itemize without change its positioning and the centering versus the items? My code below show that behaviour in fact the two item underneath have a bullet shifted down respect the other two above...

\documentclass{book}


\begin{document}

\begin{itemize}
\item One
\item Two
\end{itemize}

\renewcommand\labelitemi{\tiny$\bullet$}

\begin{itemize}
\item One
\item Two
\end{itemize}

\end{document}

Answer 1

You can raise the bullet to your liking. The following does that (to the extreme):

\documentclass{article}
\newlength{\mylen}
\setbox1=\hbox{$\bullet$}\setbox2=\hbox{\tiny$\bullet$}
\setlength{\mylen}{\dimexpr0.5\ht1-0.5\ht2}

\setlength{\parindent}{0pt}% Just for this example
\begin{document}

\begin{minipage}{0.3\linewidth}
\begin{itemize}
\item One
\item Two
\end{itemize}
\end{minipage}\begin{minipage}{0.3\linewidth}
\renewcommand\labelitemi{\tiny$\bullet$}
\begin{itemize}
\item One
\item Two
\end{itemize}
\end{minipage}\begin{minipage}{0.3\linewidth}
\renewcommand\labelitemi{\raisebox{\mylen}{\tiny$\bullet$}}
\begin{itemize}
\item One
\item Two
\end{itemize}
\end{minipage}
\end{document}

On the left is the default $\bullet$. The middle shows \tiny$bullet$ while the last shows a raised \tiny$\bullet$ to exactly the same vertical height as it should be for $\bullet$. However, as mentioned, this is a bit extreme. You could avoid all the length calculations and just use \raisebox{0.25ex}{\tiny$\bullet$} which yields a comparable positioning.

I would also suggest using enumitem which allows you to obtain a similar output using

\begin{itemize}[label=\raisebox{0.25ex}{\tiny$\bullet$}]
  %...
\end{itemize}

Answer 2

using \tiny shifts the math axis so the vertical alignment of the bullet, so what you want is not a smaller font but a smaller symbol in the same font set (or to raise the symbol by hand)

\documentclass{book}


\begin{document}

\begin{itemize}
\item One
\item Two
\end{itemize}

\renewcommand\labelitemi{$\cdot$}

\begin{itemize}
\item One
\item Two
\end{itemize}


\renewcommand\labelitemi{{\boldmath$\cdot$}}

\begin{itemize}
\item One
\item Two
\end{itemize}

\end{document}

Answer 3

The smaller symbol can be centered around the original math axis via \vcenter:

\documentclass{book}
\begin{document}

\begin{itemize}
\item One
\item Two
\end{itemize}

\renewcommand\labelitemi{$\vcenter{\hbox{\tiny$\bullet$}}$}

\begin{itemize}
\item One
\item Two
\end{itemize}

\end{document}