Quasi-toric differential inclusions

Figure 1.  (a) Polyhedral fan in two dimensions. This fan has seven cones: three two-dimensional or maximal cones, three one-dimensional cones and one zero-dimensional cone. (b) Hyperplane-generated polyhedral fan in two dimensions. This fan has 13 cones: six two-dimensional cones, six one-dimensional cones and one cone of dimension zero. (c) Polyhedral fan in three dimensions. This fan has seven cones: three three-dimensional cones, three two-dimensional cones and one cone of dimension one. (d) Hyperplane-generated polyhedral fan in three dimensions. This fan has nine cones: four three-dimensional cones, four two-dimensional cones and one cone of dimension one. Cones in (a) and (b) are pointed, while cones in (c) and (d) are not pointed. Fans in (b) and (d) are hyperplane generated, while fans in (a) and (c) are not hyperplane generated

Figure 2.  Right-hand side of a toric differential inclusion (denoted by $ F_{\mathcal{F}, \delta}( \boldsymbol{X}) $) for a hyperplane-generated fan $ \mathcal{F} $. The red region represents the set of points for which $ F_{\mathcal{F}, \delta}( \boldsymbol{X}) = \mathbb{R}^2 $. For points outside the red region, the blue cones indicate $ F_{\mathcal{F}, \delta}( \boldsymbol{X}) $, which is not $ \mathbb{R}^2 $

Figure 3.  Example of a quasi-toric differential inclusion that is not well-defined in the sense of Definition 5.2. Consider a point $ \boldsymbol{X} $ labeled by a black dot in the figure. If we iterate through the steps of Definition 5.1, we get $ dist( \boldsymbol{X}, C_1)\leq d_1 $ and $ dist( \boldsymbol{X}, \tilde{C}_1)\leq d_1 $ in Step 1. It is not clear whether $ F_{\mathcal{F}, \boldsymbol{d}}( \boldsymbol{X}) = C_1^o $ or $ F_{\mathcal{F}, \boldsymbol{d}}( \boldsymbol{X}) = {\tilde{C}_1}^o $ and hence the notion of quasi-toric differential inclusion is not well-defined for this choice of $ \boldsymbol{d} = (d_0, d_1) $

Figure 4.  Right-hand side of a quasi-toric differential inclusion (denoted by $ F_{\mathcal{F}, \boldsymbol{d}}( \boldsymbol{X}) $) for a hyperplane-generated fan $ \mathcal{F} $. The red circle represents the set of points for which $ F_{\mathcal{F}, \boldsymbol{d}}( \boldsymbol{X}) = \mathbb{R}^2 $. For points outside the red circle, the blue cones indicate $ F_{\mathcal{F}, \boldsymbol{d}}( \boldsymbol{X}) $. The numbers $ d_0, d_1 $ are chosen so that the quasi-toric differential inclusion generated by $ \mathcal{F} $ and $ \boldsymbol{d} = (d_0, d_1) $ is well-defined in the sense of Definition 5.2

Figure 5.  Two dimensional illustration of Lemma 6.3

Figure 6.  Two-dimensional illustration of Lemma 6.4

Figure 7.  (a) RHS of a toric differential inclusion (denoted by $ F_{\mathcal{F}, \delta}( \boldsymbol{X}) $) for a hyperplane-generated fan $ \mathcal{F} $. (b) RHS of a quasi-toric differential inclusion (denoted by $ F_{\mathcal{F}, { \boldsymbol{d}}}( \boldsymbol{X}) $) such that the toric differential inclusion given in part (a) can be embedded into this quasi-toric differential inclusion, i.e., $ F_{\mathcal{F}, \delta}( \boldsymbol{X})\subseteq F_{\mathcal{F}, \boldsymbol{d}}( \boldsymbol{X}) $ for every $ \boldsymbol{X}\in\mathbb{R}^n $. As in the proof of Theorem 7.3, the vector $ \boldsymbol{d} $ is constructed as follows: we set $ d_1 = \delta $ and choose $ d_0 $ large enough ($ d_0 = \lambda\alpha d_1 $) so that the quasi-toric differential inclusion is well-defined

Figure 8.  (a) RHS of a quasi-toric differential inclusion (denoted by $ F_{\mathcal{F}, \boldsymbol{d}}( \boldsymbol{X}) $) for a hyperplane-generated fan $ \mathcal{F} $. (b) RHS of a toric differential inclusion (denoted by $ F_{\mathcal{F}, \delta}( \boldsymbol{X}) $) such that the quasi-toric differential inclusion given in part (a) can be embedded into this toric differential inclusion, i.e., $ F_{\mathcal{F}, \boldsymbol{d}}( \boldsymbol{X})\subseteq F_{\mathcal{F}, \delta}( \boldsymbol{X}) $ for every $ \boldsymbol{X}\in\mathbb{R}^n $. As in the proof of Theorem 8.1, we choose $ \delta = \max(d_0, d_1) = d_0 $