Let m be a convex polygon in a plane, l its perimeter and S its area. Let M(R) be the locus of all points in the space whose distance to m is ≤R, and V(R) is the volume of the solid M(R).
a.) Prove that V(R)=34πR3+2πlR2+2SR.Hereby, we say that the distance of a point C to a figure m is ≤R if there exists a point D of the figure m such that the distance CD is ≤R. (This point D may lie on the boundary of the figure m and inside the figure.)additional question:b.) Find the area of the planar R-neighborhood of a convex or non-convex polygon m.c.) Find the volume of the R-neighborhood of a convex polyhedron, e. g. of a cube or of a tetrahedron.Note by Darij: I guess that the ''R-neighborhood'' of a figure is defined as the locus of all points whose distance to the figure is ≤R. geometryperimeter3D geometryspherecombinatorial geometryIMO ShortlistIMO Longlist