## 2 Problem Formulation
\begin{split}& h(\mathbf{u},\mathcal{X}+\mathcal{Y})=h(\mathbf{ u},\mathcal{X})+h(\mathbf{u},\mathcal{Y})\quad\forall\mathbf{u}\in\mathbb{R}^{s},\\ &\mathcal{X}\subset\mathcal{Y}\ \ \Leftrightarrow\ \ h(\mathbf{u},\mathcal{X})\leq h(\mathbf{u},\mathcal{Y})\quad\forall \mathbf{u}\in\mathbb{R}^{s},\end{split}  \tag{8}\]  
where \(h(\mathbf{v},\mathcal{Z})\) is a support function of set \(\mathcal{Z}\) in direction \(\mathbf{v}\). Based on the inclusion \(\{\mathbf{0}\}\subset\mathcal{V}(\mathbf{x})\) and the property (8) it is possible to express the solution set \(\Xi_{\mathrm{uni}}\) in the form of set of inequalities. The set \(\mathcal{V}(\mathbf{x})\) is a zonotope, so a finite set \(\mathcal{U}\subset\mathbb{R}^{s}\) exist, such that  
\[\{\mathbf{0}\}\subset\mathcal{V}(\mathbf{x})\ \ \Leftrightarrow\ \ h\left(\mathbf{u},\{ \emptyset\}\right)=0\leq h\left(\mathbf{u},\mathcal{V}(\mathbf{x})\right)\quad \text{for}\quad\mathbf{u}\in\mathcal{U}. \tag{9}\]  
It is relevant, that if there is equivalence (9) for set \(\mathcal{U}=\overline{\mathcal{U}}\), then the equivalence is satisfied for all \(\mathcal{U}=\overline{\overline{\mathcal{U}}}\), such that \(\overline{\overline{\mathcal{U}}}\supset\overline{\mathcal{U}}\). It should be noted that each convex set can be described with a finite or infinite set of hyperplanes corresponding to successive \(\mathbf{u}_{i}\) vectors that are normal to them. However, in the case of a zonotope, due to the fact that it is a polyhedron and its symmetry, it is possible to limit this set of vectors to a finite set. The equivalences (6) and (9) lead to the description of solution set \(\Xi_{\mathrm{uni}}\) in the form of set of inequalities  
\[\mathbf{x}\in\Xi_{\mathrm{uni}}\ \ \Leftrightarrow\ \ 0\leq h\left(\mathbf{u}, \mathcal{V}(\mathbf{x})\right)\quad\text{dla}\quad\mathbf{u}\in\mathcal{U}. \tag{10}\]  
where \(\mathcal{U}\) is a set, for which the equivalence (9) is satisfied. Determining the above description comes down to determining the set \(\mathcal{U}\) and calculation of \(h\left(\mathbf{u},\mathcal{V}(\mathbf{x})\right)\). Let \(\overline{F}\subset\mathbb{R}^{s}\) be the set of non-zero vectors, satisfying the following conditions: (a) for each zonotope generator \(\mathcal{V}(\mathbf{x})\) set \(\overline{F}\) contains non-zero vector parallel to him, (b) \(\overline{F}\) contains \(s\) linearly independent vectors. Moreover, let \(\mathcal{F}\) stand for the family of all subsets \(\overline{F}\), which consist of \(s-1\) linearly independent vectors. **Proposition 1**. : _Let \(\mathcal{U}^{+}\subset\mathbb{R}^{s}\) be the set satisfying the following condition: for each \(F\in\mathcal{F}\) exists the non-zero vector \(\mathbf{u}\in\mathcal{U}^{+}\), such that \(\langle\mathbf{u},F\rangle=0\). Then for \(\mathcal{U}=\mathcal{U}^{+}\cup(-\mathcal{U}^{+})\) there is equivalence in (9)._  
The above proposition and the equivalence in (10) lead to the description  
\[\mathbf{x}\in\Xi_{\mathrm{uni}}\ \ \Leftrightarrow\ \ (0\leq h\left(\mathbf{u}, \mathcal{V}(\mathbf{x})\right)\ \ \wedge\ \ 0\leq h\left(-\mathbf{u},\mathcal{V}(\mathbf{x})\right))\quad\text{dla}\quad \mathbf{u}\in\mathcal{U}^{+}. \tag{11}\]Values \(h\left(\mathbf{u},\mathcal{V}(\mathbf{x})\right)\) and \(h\left(-\mathbf{u},\mathcal{V}(\mathbf{x})\right)\) remain to be calculated. The method of determining the set \(\mathcal{U}^{+}\) is presented in Section 3.