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 $M\left( R\right) .$


[i]a.)[/i] 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:

[i]b.)[/i] Find the area of the planar $R$-neighborhood of a convex or non-convex polygon $m.$

[i]c.)[/i] Find the volume of the $R$-neighborhood of a convex polyhedron, e. g. of a cube or of a tetrahedron.

[b]Note by Darij:[/b] I guess that the ''$R$-neighborhood'' of a figure is defined as the locus of all points whose distance to the figure is $\leq R.$
### Part (a)

To prove that the volume \( V(R) \) of the solid \( M(R) \) is given by:
\[ V(R) = \frac{4}{3} \pi R^3 + \frac{\pi}{2} l R^2 + 2SR, \]
we need to consider the contributions to the volume from different parts of the solid.

1. **Volume of the Prism:**
   The polygon \( m \) has an area \( S \). If we extend this polygon vertically by a distance \( 2R \) (i.e., \( R \) above and \( R \) below the plane of the polygon), we form a prism with height \( 2R \) and base area \( S \). The volume of this prism is:
   \[
   V_{\text{prism}} = S \cdot 2R = 2SR.
   \]

2. **Volume of the Half-Cylinders:**
   Each edge of the polygon \( m \) can be thought of as the base of a half-cylinder with radius \( R \) and length equal to the length of the edge. The total length of the edges of the polygon is the perimeter \( l \). The volume of a single half-cylinder with base length \( a_i \) and radius \( R \) is:
   \[
   V_{\text{half-cylinder}} = \frac{1}{2} \pi R^2 a_i.
   \]
   Summing over all edges, the total volume of the half-cylinders is:
   \[
   V_{\text{half-cylinders}} = \sum \frac{1}{2} \pi R^2 a_i = \frac{\pi}{2} R^2 \sum a_i = \frac{\pi}{2} R^2 l.
   \]

3. **Volume of the Spherical Caps:**
   At each vertex of the polygon, there is a spherical cap of radius \( R \). The volume of a single spherical cap can be considered as part of a sphere of radius \( R \). The total volume of these spherical caps, when combined, forms a complete sphere of radius \( R \). The volume of a sphere of radius \( R \) is:
   \[
   V_{\text{sphere}} = \frac{4}{3} \pi R^3.
   \]

Adding these contributions together, we get the total volume \( V(R) \):
\[
V(R) = V_{\text{prism}} + V_{\text{half-cylinders}} + V_{\text{sphere}} = 2SR + \frac{\pi}{2} R^2 l + \frac{4}{3} \pi R^3.
\]

Thus, we have:
\[
V(R) = \frac{4}{3} \pi R^3 + \frac{\pi}{2} l R^2 + 2SR.
\]

### Part (b)

To find the area of the planar \( R \)-neighborhood of a convex or non-convex polygon \( m \):

1. **Area of the Polygon:**
   The area of the polygon itself is \( S \).

2. **Area of the Half-Cylinders:**
   The perimeter of the polygon is \( l \). The area of the half-cylinders around the edges is:
   \[
   A_{\text{half-cylinders}} = l \cdot \pi R.
   \]

3. **Area of the Circular Regions:**
   At each vertex, there is a circular region of radius \( R \). The total area of these circular regions, when combined, forms a complete circle of radius \( R \). The area of a circle of radius \( R \) is:
   \[
   A_{\text{circle}} = \pi R^2.
   \]

Adding these contributions together, we get the total area \( A(R) \):
\[
A(R) = 2S + l \pi R + 4 \pi R^2.
\]

### Part (c)

To find the volume of the \( R \)-neighborhood of a convex polyhedron (e.g., a cube or a tetrahedron):

1. **Volume of the Polyhedron:**
   Let the volume of the polyhedron be \( V \).

2. **Volume of the Prisms:**
   Each face of the polyhedron can be extended by a distance \( R \) to form a prism. The total volume of these prisms is the sum of the areas of the faces times \( R \).

3. **Volume of the Cylindrical Regions:**
   Each edge of the polyhedron can be extended by a distance \( R \) to form a cylindrical region. The total volume of these cylindrical regions is the sum of the lengths of the edges times \( \pi R^2 \).

4. **Volume of the Spherical Caps:**
   At each vertex of the polyhedron, there is a spherical cap of radius \( R \). The total volume of these spherical caps, when combined, forms a complete sphere of radius \( R \).

Combining these contributions, we get the total volume \( V(R) \) for a convex polyhedron.

The final answer is \( \boxed{ V(R) = \frac{4}{3} \pi R^3 + \frac{\pi}{2} l R^2 + 2SR } \).