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Let $S$ be a smooth compact surface in $\mathbb{R}^3$, with two distinct, distinguished points $a,b \in S$. Let $R(a)$ be all the points of $S$ closer to $a$ than to $b$, and $R(b)$ all the points of $S$ closer to $b$ than to $a$, where distance is measured by shortest paths on the surface. The points equally distant from $a$ and $b$ form a bisector, a closed curve on $S$. This is in a sense the Voronoi diagram of the two points on $S$.

Q1. Is the area of $R(a)$ equal to the area of $R(b)$? Suppose $S$ has genus zero, or is even convex—Do these restrictions change the answer?

Answer: Jack Huizenga's examples in his comment show the answer is a strong No, even for a convex surface.

Q2. Following Vitali Kapovitch's comment suggestion, is the sphere the only closed, bounded, genus-zero surface for which $R(a)=R(b)$ for every pair $a,b$?

Addendum. There is no need for $S$ to be smooth, and indeed I am primarily interested in polyhedra, specifically, convex polyhedra. I am now thinking it is an interesting question to identify all the pairs of points $(x,y)$ for which their bisector $B=B(x,y)$ halves the surface area, e.g.:
            Box bisector
But I should save that for perhaps a separate question...

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    $\begingroup$ Join two spheres by a small cylinder, where one of the spheres is much larger than the other. Take $a$ to be on the cylinder, and $b$ to be on the smaller sphere, away from the cylinder. Clearly $R(a)$ is much larger than $R(b)$. Simply taking the surface to be a very long narrow cylinder (smoothed out accordingly) and picking $a$ and $b$ appropriately, one can rig convex examples. $\endgroup$
    – Jack Huizenga
    Jan 18, 2012 at 1:19
  • $\begingroup$ @Jack: Ah, I see, very nice example--Thanks! $\endgroup$
    – Joseph O'Rourke
    Jan 18, 2012 at 1:26
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    $\begingroup$ I would guess that pretty much the opposite should be true. that is if the areas are equal for any $a$ and $b$ then S should have constant curvature. $\endgroup$
    – Vitali Kapovitch
    Jan 18, 2012 at 1:58
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    $\begingroup$ Indeed, Jack's example suggests another natural class if we start with a cone and place b at the apex and a not too far from b. Gerhard "Ask Me About System Design" Paseman, 2012.01.17 $\endgroup$
    – Gerhard Paseman
    Jan 18, 2012 at 2:53
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    $\begingroup$ For question 2, I nominate without proof the torus. Gerhard "Ask Me About System Design" Paseman, 2012.01.18 $\endgroup$
    – Gerhard Paseman
    Jan 18, 2012 at 15:43

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