Let
be a convex polygon in a plane,
its perimeter and
its area. Let
be the locus of all points in the space whose distance to
is
and
is the volume of the solid ![M\left( R\right) .](/media/m/d/2/e/d2eeefbcaef0dcc533603be7d0854ab5.png)
a.) Prove that![V (R) = \frac 43 \pi R^3 +\frac{\pi}{2} lR^2 +2SR.](/media/m/9/8/1/981e0b65a04721d82e0d96e31b1f1985.png)
Hereby, we say that the distance of a point
to a figure
is
if there exists a point
of the figure
such that the distance
is
(This point
may lie on the boundary of the figure
and inside the figure.)
additional question:
b.) Find the area of the planar
-neighborhood of a convex or non-convex polygon ![m.](/media/m/c/6/9/c69a166e977669b9486ba58ac831eea1.png)
c.) Find the volume of the
-neighborhood of a convex polyhedron, e. g. of a cube or of a tetrahedron.
Note by Darij: I guess that the ''
-neighborhood'' of a figure is defined as the locus of all points whose distance to the figure is
![m](/media/m/1/3/6/1361d4850444c055a8a322281f279b39.png)
![l](/media/m/e/e/9/ee975101080f37986f56baaf4c3cdcd2.png)
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
![M\left( R\right)](/media/m/2/a/f/2afa6d0ba232f9df018458227e0a76e8.png)
![m](/media/m/1/3/6/1361d4850444c055a8a322281f279b39.png)
![\leq R,](/media/m/f/a/c/fac908aecf00ba5c22b23ae359160b1a.png)
![V\left(R\right)](/media/m/d/2/9/d2987acca5e6a169ad9c0268782b374c.png)
![M\left( R\right) .](/media/m/d/2/e/d2eeefbcaef0dcc533603be7d0854ab5.png)
a.) Prove that
![V (R) = \frac 43 \pi R^3 +\frac{\pi}{2} lR^2 +2SR.](/media/m/9/8/1/981e0b65a04721d82e0d96e31b1f1985.png)
Hereby, we say that the distance of a point
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
![m](/media/m/1/3/6/1361d4850444c055a8a322281f279b39.png)
![\leq R](/media/m/0/3/3/033f6d4543e7b655732f15707ba4eeec.png)
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
![m](/media/m/1/3/6/1361d4850444c055a8a322281f279b39.png)
![CD](/media/m/8/9/5/895081147290365ccae028796608097d.png)
![\leq R.](/media/m/9/7/b/97ba4924f96b7b1f66492cd627265861.png)
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
![m](/media/m/1/3/6/1361d4850444c055a8a322281f279b39.png)
additional question:
b.) Find the area of the planar
![R](/media/m/4/d/7/4d76ce566584cfe8ff88e5f3e8b8e823.png)
![m.](/media/m/c/6/9/c69a166e977669b9486ba58ac831eea1.png)
c.) Find the volume of the
![R](/media/m/4/d/7/4d76ce566584cfe8ff88e5f3e8b8e823.png)
Note by Darij: I guess that the ''
![R](/media/m/4/d/7/4d76ce566584cfe8ff88e5f3e8b8e823.png)
![\leq R.](/media/m/9/7/b/97ba4924f96b7b1f66492cd627265861.png)