This is my interpretation of andy's example in his comment to Igor's answer.
$s$ and $t$ are the green points.
Imagine the outer face red because it contains a (green) point.
$L$ and $M$ are blue because they contain no points.
Separating Arrangement http://people.csail.mit.edu/%7Eorourke/MathOverflow/SeparatingArrangement.jpg
Removing $e$ would lead to a less efficient separator.
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This is my interpretation of andy's example in his comment to Igor's answer.
$s$ and $t$ are the green points.
Imagine the outer face red because it contains a (green) point.
$L$ and $M$ are blue because they contain no points.
Separating Arrangement http://people.csail.mit.edu/%7Eorourke/MathOverflow/SeparatingArrangement.jpg
Removing $e$ would lead to a less efficient separator.
This is my interpretation of andy's example in his comment to Igor's answer.
$s$ and $t$ are the green points.
Imagine the outer face red because it contains a (green) point.
$L$ and $M$ are blue because they contain no points.
Removing $e$ would lead to a less efficient separator.
