I've not yet convinced myself that this is soluble in the general case.

In the 2D case, if all the points in both groups have one coordinate in common, and there are an odd number of points in each group or in the more general case of all the points lying on a straight line at any arbitrary angle.: 

+-----------+   +-----------+    +-----------+
|     .     |   |           |    |      .    |
|     x     |   |           |    |     .     |
|     x     |   |           |    |    .      |
|     .     |   |.xx . x .  |    |   x       |
|     .     |   |           |    |  x        |
|     x     |   |           |    | x         |
+-----------+   +-----------+    +-----------+

[download]

Unless you consider the line passing through all the points satisfies the criteria of having an equal number of each type of point on either side; ie. none?

Examine what is said, not who speaks -- Silence betokens consent -- Love the truth but pardon error.
Lingua non convalesco, consenesco et abolesco. -- Rule 1 has a caveat! -- Who broke the cabal?
"Science is about questioning the status quo. Questioning authority".
In the absence of evidence, opinion is indistinguishable from prejudice.

In reply to Re^3: Puzzle: The Ham Cheese Sandwich cut. by BrowserUk
in thread Puzzle: The Ham Cheese Sandwich cut. by Perl Mouse

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