The screenshot shows two polytopes and its Minkowski sum and is obtained by the following commands:

P=cube(3); Q=ball(3); R=P+Q; plot(P); plot(Q); plot(R);

There are different options to enter a convex polyhedron:

**V-representation:** Finitely many points, for
example \(0 \choose 3\), \(3 \choose 0\), \(2 \choose 2\), and
finitely many directions, for example \(-1 \choose 0\), \(0 \choose -1\),
are stored column-wise, respectively,
in a field V and a field D of a structure rep:

rep.V=[0 3; 3 0; 2 2]'; rep.D=[-1 0;0 -1];

A polyhedron P is defined, where option 'v' indicates that rep is a V-represenation. The result is plotted: P=polyh(rep,'v'); plot(P); |

**H-representation:** A polyhedron P of the form
\[ P=\{x \in \mathbb{R}^n |\; a \leq Bx \leq b,\; l \leq x \leq u\}\]
is considered. For example, an H-representation for
\[ P=\{x \in \mathbb{R}^2 |\; 2 \leq 2 x_1 + x_2,\; 2 \leq x_1 + 2 x_2,\, x_1 \geq 0\}\]
can be enererd as

clear rep; rep.B=[2 1; 1 2]; rep.a=[2; 2]; rep.l=[0; -Inf];

A polyhedron Q is defined, where option 'h' indicates that rep is an H-represenation. The result is plotted: Q=polyh(rep,'h'); plot(Q); |

R=P&Q; plot(R); |

The Cartesian product of P and Q is computed and stored in S. The image of the 4-dimensional polyhedron S under the (\(3 \times 4\))-matrix M is a 3-dimensional polyhedron T with 5 vertices and 4 extremal directions. The length of the unbounded edges in the plot is set to 10: S=P:Q; M=[2 1 3 8; 1 6 8 5; 1 7 5 0]; T=S.im(M); opt.dirlength=10; plot(T,opt); |