The following is from Prof. Jahn's book " Intro. to the theory of nonlinear optimization" about tangent cone:
After definition, he gave an example:
My question is: Does the tangent cone include S? (In the def., it says tangent cone is a set of all tangent vectors to S at bar(x))
In the figure, T(S, bar(x)) should not include the region S?
On
$T(S, \bar{x}) + \bar{x}$ includes $S$ if $S$ is convex. Note that it is also a cone, i.e. if $y \in T(S, \bar{x})$, then $\lambda y \in T(S, \bar{x})$ for any real $\lambda \geq 0$.
To see the first assertion, for any $x \in S$ take a sequence $(x_n)$ of points on the segment $\{(1-\alpha) x + \alpha\bar{x}: \alpha \in [0, 1)\}$ that converges to $\bar{x}$. Since $S$ is convex, this segment is contained in $S$. If $x_n = (1-\alpha_n) x + \alpha_n\bar{x}$, take $\lambda_n = 1/(1-\alpha_n)$, and you have $\lambda_n(x_n - \bar{x}) = x - \bar{x}$ for all $n$.
First of all, let's clear something up. Note that the tangent cone is a set of vectors at a point, which is not exactly the set of points in the highlighted region (the two sets contain elements of different "data types"). The question that you seem to be thinking of, then, is whether the tangent cone at a point contains vectors that point from $\bar x$ into $S$.
Note that if your point $\bar x$ happens to be $\bar 0$, then you can validly think of the vectors emerging from $\bar x$ as points whose coordinates are given by their components, hence the first picture.
In general, the tangent cone (at a point $\bar x \in S$) may or may not contain vectors that, when drawn emerging from $\bar x$, pass through $S$. In your particular example, the tangent cone does contain such vectors. In general, as long as $S$ is convex, you can think of the tangent cone as comprising (of the closure) of "all the directions that take you through $S$".
As an example of when the tangent cone doesn't necessarily contain vectors pointing into the space, take your region to be the border of the set $S$ that is drawn.