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Soviet Union 2

Source: IMO LongList 1959-1966 Problem 6

September 1, 2004
geometryperimeter3D geometryspherecombinatorial geometryIMO ShortlistIMO Longlist

Problem Statement

Let mm be a convex polygon in a plane, ll its perimeter and SS its area. Let M(R)M\left( R\right) be the locus of all points in the space whose distance to mm is ≤R,\leq R, and V(R)V\left(R\right) is the volume of the solid M(R).M\left( R\right) .
a.) Prove that V(R)=43πR3+π2lR2+2SR.V (R) = \frac 43 \pi R^3 +\frac{\pi}{2} lR^2 +2SR.
Hereby, we say that the distance of a point CC to a figure mm is ≤R\leq R if there exists a point DD of the figure mm such that the distance CDCD is ≤R.\leq R. (This point DD may lie on the boundary of the figure mm and inside the figure.)
additional question:
b.) Find the area of the planar RR-neighborhood of a convex or non-convex polygon m.m.
c.) Find the volume of the RR-neighborhood of a convex polyhedron, e. g. of a cube or of a tetrahedron.
Note by Darij: I guess that the ''RR-neighborhood'' of a figure is defined as the locus of all points whose distance to the figure is ≤R.\leq R.
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