Školjka

%V0 Let $m$ be a convex polygon in a plane, $l$ its perimeter and $S$ its area. Let $M\left( R\right)$ be the locus of all points in the space whose distance to $m$ is $\leq R,$ and $V\left(R\right)$ is the volume of the solid $M\left( R\right) .$ a.) Prove that $$V (R) = \frac 43 \pi R^3 +\frac{\pi}{2} lR^2 +2SR.$$ Hereby, we say that the distance of a point $C$ to a figure $m$ is $\leq R$ if there exists a point $D$ of the figure $m$ such that the distance $CD$ is $\leq R.$ (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 $\leq R.$