# A higher-order analogy of the fact that a set is star-convex with vantage point at every point in it iff it is convex.

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.)