An algorithm that find the max X/Y in a polygon in O(log n)
Assume the polygon is a convex polygon; otherwise there might not be an $O(\log n)$ algorithm to find the maximum of X-coordinates or Y-coordinates.

First, let us handle the case of X-coordinate.

Let $x_1, x_2, \cdots, x_n$ be the X-coordinate of the points. Since $x_1$ is the minimum X-coordinate, that sequence is a bitonic sequence. Here are two nice answers to how to sort a bitonic sequence.

Next, let us solve the case of Y-coordinate, which is mildly more complicated.

Let $y_1, y_2, \cdots, y_n, y_{n+1}=y_1$ be the Y-coordinate of the points. We can drill down into the following cases by comparing $y_1$ against $y_2$ and against $y_n$.


$C_{\lt, \lt}$ when $y_1\lt y_2$ and $y_1\lt y_n$. Here $y_1$ is the minimum.
$C_{\gt, \gt}$ when $y_1\gt y_2$ and $y_1\gt y_n$. Here $y_1$ is the maximum.
$C_{\lt, \gt}$ when $y_1\lt y_2$ and $y_1\gt y_n$. Here Y-coordinates goes up from $y_1$ to the maximum, then goes down to the minimum, then goes up to $y_1$.
$C_{\gt, \lt}$ when $y_1\gt y_2$ and $y_1\lt y_n$. Here Y-coordinates goes down from $y_1$ to the minimum, then goes up to the maximum, then goes down to $y_1$.


The first two cases are bitonic sequences, which can be treated as before.  

Let us check the case of $C_{\lt, \gt}$; the case $C_{\gt, \lt}$ can be handled similarly. 

Let $y_M$ is the maximum and $y_m$ is the minimum, where $M&lt;m$. Given an index $1&lt;i&lt;n$, we can compare $i$ to $m$ and $M$ as follows:


If $y[i-1] &lt; y[i] &lt; y[i+1]$ and $y[i] &gt; y_1$, then $i &lt; M$.
If $y[i-1] &lt; y[i] &gt; y[i+1]$, then $i = M$.
If $y[i-1] &gt; y[i] &gt; y[i+1]$, then $M &lt; i &lt; m$.
If $y[i-1] &gt; y[i] &lt; y[i+1]$, then $i = m$.
If $y[i-1] &lt; y[i] &lt; y[i+1]$ and $y[i] &lt; y_1$, then $i &gt; m$.


The above result enables us to cut the search intervals in half each time with 2 or 3 comparisons, thus ensuring an algorithm with $O(\log n)$ time-complexity.



Exercise. What about the case when we have a sequence of $n$ distinct numbers, in which there are $t$ local maximums and $t+1$ local minimums?