Vol. 16, No. 1, 1966

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A theorem on partitions of mass-distribution

V. V. Menon

Vol. 16 (1966), No. 1, 133–137

A ‘bisector’ of a continuous mass-distribution M in a bounded region on the plane is defined as a straight line such that the two half-planes determined by this line contain half the mass of M each. It is known that there exists at least one point (in the plane) through which pass three bisectors of M.

Theorem. Let, for a continuous mass distribution M, the point P through which three bisectors pass be unique. Then all bisectors of M pass through p.

The following corollary also is established: For a convex figure K (i.e., compact convex set with nonempty interior) to be centrally symmetric, it is necessary and sufficient that the point through which three bisectors of area pass be unique.

Mathematical Subject Classification
Primary: 52.40
Received: 25 January 1964
Revised: 30 July 1964
Published: 1 January 1966
V. V. Menon