I am reading the paper Positive Invariance Condition for Continuous Dynamical Systems Based on Nagumo Theorem, and specifically I am concerning Theorem 3.1, where the tangent cone at boundary points of polyhedron is described.
I am wondering whether there is a typo in the theorem? That, $\{y\in\mathbb{R}^n|g_{i_j}^Ty\leq 0,j=1,2,\cdots,k\}$ should be $\{y\in\mathbb{R}^n|g_{i_j}^Ty\leq b_{i_j},j=1,2,\cdots,k\}$
So far I don't quite understand these concepts thus it is not easy for me to see whether this is a typo or not. It would be very appreciate if someone can help me out.
Given a closed convex set $K\subset \mathbb{R}^n$ and a point $x\in K$ the tangent cone to $K$ at $x$ is defined by \begin{equation} T_K(x):=\text{cl}\{v\in \mathbb{R}^n\;\vert\;\exists \lambda \geq 0 \text{ and } \exists y \in K\text{ such that }v=\lambda (y-x)\}. \end{equation} (here, $\text{cl}$ denotes the closure, this is one possible definition, many others can be found, for more general classes of sets). Some general facts: if $x\in \text{Int}(K)$ then $T_K(x)=\mathbb{R}^n$, $T_{\{a\}}(a)=\{0\}$,....
More importantly, $T_K(x)$ is a cone ($v\in T_K(x) \Leftrightarrow \lambda v\in T_K(x) ,\forall \lambda \geq 0$), and it is a convex set. Informally and very roughly speaking, it can be seen as the (closure) of the set of feasible directions at $x$, i.e. the direction that, starting at $x$, I can follow without escaping $K$, at least for small steps.
Now, in your case of a polyhedral set, I give a hint: at each point of the boundary you should look which constraints of the form $g_i^\top x\leq b_i$ are active since, the feasible directions...