Question

(a) Have C be a circular region in R^2. Determine the normal cone for a point on its circumference.
(b) Have C be a rectangular region in R^2. Determine the normal cone for a point on its boundary.

Answer 1

a)

By definition, normal cone for a set $X$ at point $x$ is defined in such a way:

$$N_X(x) = \{z \mid \exists x_k \subset X, \{z_k\} \text{ s.t. } z_k \in T_X(x_k)^*, x_k \to x, z_k \to z\}$$

Here $T_X(x)$ is the tangent cone to the set $X$ at point $x$. It is defined in such a way:

$$T_X(x) = \{0\} \cup \left\{y \mid y \neq 0, \exists \{x_k\} \subset X \text{ s.t. } x_k \neq x \text{ and } x_k \to x, \frac{x_k - x}{\|x_k - x\|} \to \frac{y_k}{\|y\|} \right\}$$

Without losing generality, let $x$ be a point at the origin and circular region $C$ lies in the upper half-plane. Since the tangent cone equals to

$T_X(x) = \{(x,y) \mid y > 0\}$ the normal cone equals to

$$N_X(x) = \{(x, y) \mid y = 0\}$$

(b)

In case $C$ is rectangular region in $R^2$, $x$ is point on the boundary, then as in (a)

$$T_X(x) = \{(x, y) \mid y > 0\}$$ $$N_X(x) = \{(x, y) \mid y = 0\}$$