common intersection of circles

[Jie Lin]

Hello computational geometry people,

I have n points {z_1, z_2, ... z_n} in the plane, located with radius R from
the origin. Denote H the convex hull of these n points plus the origin.
Denote C_i the circle with radius R centered at each point z_i. Denote S the
intersection of C_i.

Now, it can be shown that if the origin 0 is a corner of the convex hull H,
then S has non-empty interior (contains more than point 0).

My question is that whether or not the centroid (center of mass) of S ---the
common intersection of circles, is located in the convex hull H. If not,
does the line segment from 0 to the centroid of S has a positive
intersection with H?

My feeling is yes, but I had trouble showing it. I hope someone in this
group can help me out here. Thanks in advance.

Sincerely,

Jie


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