There is a simple relation between star-convex sets and convex sets:
A set $S$ is convex iff it is star-convex with vantage point at every point in it.
Proof.
- $\Longrightarrow \ $ if $S$ is convex, then for every $s_0,s\in S$ the line between $s_0$ and $s$ is in $S$.
- $\Longleftarrow \ $ if $S$ is star-convex at every $s_0\in S$, then for any $a,b\in S$ the line between $a$ and $b$ is in $S$ because $a$ is a vantage point of $S$.
Thus
Convexity is the property characterizing any set $S$ that is star-convex with vantage point at every point in $S$.
When contemplating about a higher-order analogy to star-convex sets I came up with the following definition:
A set $S\subset \Bbb R^2$ is second-order star-convex at $s_0\in \mathop{cl}(S)$ and $\bar s_0 \in \mathop{cl}(\Bbb R^2 \setminus S)$ iff $S$ has at most one "hole" along any cubic $C$ passing the points $s_0$ and $\bar s_0$. In other terms, $S \cap C$ consists of at most two connected sets.
Note: Closure and connectedness are meant with respect to the homogenous coordinates, so that two branches of a hyperbola form one connected set (as they share a point at infinity), and $s_0$ can be a point at infinity if points of $S$ represented in homogenous coordinates converge to it; and analogically $\bar s_0$ can be a point at infinity if points of $\Bbb R^2 \setminus S$ represented in homogenous coordinates converge to it.
Example: Putting $s_0$ and $\bar s_0$ to be both the point $(0,1,0)$ at infinity (represented in homogenous coordinates), then each conic $C\ni x_0,\bar x_0$ would be a vertically-symmetric parabola (or a line), and the condition would characterize epigraphs of functions with positive/negative third-order derivative. (To distinguish between the epigraphs $\mathop{epi}(f)$ of functions with $f^{(3)}\geq 0$ from those with $f^{(3)}\leq 0$ we would need the point $s_0$ to be left from $\bar s_0$, which I don't know how to specify using the homogenous coordinates.)
Question:
Can you characterize all the sets $S \subset \mathbb R^2$ such that $S$ is second-order star-convex at all $s_0\in \mathop{cl}(S)$ and $\bar s_0 \in \mathop{cl}(\Bbb R^2\setminus S)$?
(If it was easier / more interesting to consider $s_0\in S$ and $\bar s_0 \in \Bbb R^2\setminus S$ instead then feel free to do so.)
