I am new to the convex geometry. I recently came across a term affine convex cone. Wikipedia mentions that
"An affine convex cone is the set resulting from applying an affine transformation to a convex cone. A common example is translating a convex cone by a point $p: p+C.$ Technically, such transformations can produce non-cones. For example, unless $p=0$, $p+C$ is not a linear cone. However, it is still called an affine convex cone."
My questions are:
$(1)$ Given any compact convex set $A\subset \mathbb{R}^2$ and a point $b\notin A$, can we always draw two lines from $b$ that support $A$?
$(2)$ Is it possible to draw more than two lines from $b$ that support $A$?
$(3)$ If two lines (assuming we can draw exactly two lines) are $L_1$ and $L_2$. Then is the convex hull of $L_1$ and $L_2$ an affine convex cone at $b$?
EDIT Suppose that $A$ contains at least two points such that they do not lie on a line passing through origin at the same time.
Thanks in advance.