How to center an `itemize` in beamer, together with its headline?

Here is one way; using a tabular-like structure:

\documentclass{beamer}
\usetheme{CambridgeUS}
\newcommand{\tabitem}{%
  \usebeamertemplate{itemize item}\hspace*{\labelsep}}

\begin{document}

\begin{frame}{Centering the itemize}
  My first try: Centering the itemize
  \begin{itemize}
    \item item1
    \item item2
  \end{itemize}

  \begin{center}
    \begin{tabular}{@{}l@{}}
      My second try: Centering the itemize \\
      \tabitem item1 \\
      \tabitem item2
    \end{tabular}
  \end{center}
\end{frame}

\end{document}

You insert an "\item" inside the tabular using \tabitem.

Another way is to include the itemize list within the minipage environment and center the minipage. The key is to make the width of the minipage just enough to hold the widest item in the list. I use trial and error, but I am sure there is a more cleaver way.

\documentclass{beamer}
\usetheme{CambridgeUS}

\begin{document}

\begin{frame}{Centering the itemize}
  \begin{center}
    \begin{minipage}{0.14\textwidth}
      \begin{itemize}
        \item item1
        \item item2
      \end{itemize}
    \end{minipage}
  \end{center}
\end{frame}

\end{document}

There is another way:

\begin{itemize}
    \item  \begin{center} item1 \end{center}
    \item  \begin{center} item2 \end{center}
\end{itemize}