Below are images for the examples from the statement. The dashed line indicates the edges of the new convex hull, the solid line indicates the edges of the old convex hull and the common edges. The blue arrow shows the optimal movement of the white point.

\begin{center}
\includegraphics[scale=0.9]{drawH.1}

\small{Example 1}
\end{center}

\begin{center}
\includegraphics[scale=0.6]{drawH.2}

\small{Example 2}
\end{center}

\begin{center}
\includegraphics[scale=0.6]{drawH.3}

\small{Example 3}
\end{center}