For a set $C$ (which may not be convex) and a point $x\in C$:
- Bouligand's Tangent cone is defined as $$ T(C,x) = \left\{v : \lim_{\theta\to 0_+} \inf \frac{d(x+\theta v, C)}{\theta} = 0\right\} $$ and where $d(x,C) = \min_{y\in C} \|x-y\|$ the distance from a point to a set.
- Clarke's tangent cone is $$ T_C(C,x) = \left\{ v : \lim_{y\to x, y\in C, \theta\to 0_+} \frac{d(y+\theta v,C)}{\theta} = 0 \right\} $$
- A set is regular if $T(x,C) = T_C(x,C)$ for all $x\in C$.
My questions are a bit general, as I'm trying to build intuition.
- If a set is convex, then is it always regular? (Including possibly infinitely-dimensional sets? What if we restrict to finite-dimensional sets?) Would it be fair to say that here, both definitions boil down to the "usual" tangent cone definition, e.g. $$ T_0(C,x) = \lim_{r\to 0}\mathrm{cone}(\{y\in C: \|x-y\|\leq r\}) $$
- Do funny things happen if $C$ is a low dimensional subspace (e.g. convex but unbounded and with empty interior?)
- Now assume that I have a set which is nonconvex, shaped like a cashew (e.g. no nonsmooth points.) Then it seems like the tangent cone at any point is just a halfspace, using either definition. Does this seem true?
- Now assume that I have a set which is "pointy" and nonconvex, like Pacman. In particular, take $x$ to be the point most inside Pacman's mouth. More precisely, consider $$ C = \{x : \|x\| \leq 1\} \cap \{x : \angle(x_2,x_1) > \alpha \text{ or }\angle (x_2,x_1) < \alpha\} $$for some $\pi/2 > \alpha > 0$, and take $x = 0$. I suppose the tangent cone, using either definition, at this point, is the set $\{x : \angle(x_2,x_1) > \alpha \text{ or }\angle (x_2,x_1) < \alpha\}$, and the normal cone, defined as the polar of the tangent cone, is empty (in both definitions). Does this sound sensible?
- Finally, the main question is: what is an example of a set which is not regular? I presume such sets must be nonconvex; can they also be finite-dimensional? What about compact / closed / bounded?
Thanks for any discussion!
