# Generalization of notion of convexity

I am searching for the correct term for the following, if it exists.

A set $X\subset \mathbb{R}^2$ is called $r$-convex if for any two points $x_1, x_2\in X$ such that there exists an arc of radius $r$ connecting these two points, at least one arc lies in the set $X$.

Note 1: Obviously, there are always two arcs of radius $r$ (if one exists) connecting two points.

Note 2: When $r\rightarrow +\infty$ we have the definition of convex set.

My main interest is efficient algorithms for computation of convex hulls based on this definition of convexity.

• I think I would disagree with your first note as, surely, if we take the set X to be a semicircle of radius r then there is only one arc of radius r between any two points. – Stijn Apr 18 '14 at 12:56
• @Stijn, I think he's saying that there are two arcs in $\mathbb{R}^2$, not necessarily in $X$. – Dylan Thurston Apr 18 '14 at 14:20
• It seems like a weird notion. Where does it come up? – Dylan Thurston Apr 18 '14 at 14:20
• Yes, I am talking about two arcs in $\mathbb{R}^2$. – Fedor Nikitin Apr 18 '14 at 14:27
• @Dylan At the moment I am working on algorithm for rounding sharp corners in polygon. Let $P$ be a set of points lying in polygon. I need to find $P^\prime$ such that its border consists of segments and arcs. Arc radius should not exceed $r$ and there are no sharp angles (less than $\pi$). I have guess that $P\prime = \mathbb{R}^2 \ co_r (\mathbb{R}^2 \ P)$, where $co_r$ is convex hull in the sense of definition above. My idea is to apply algorithms for regular convex hull with modified definition of convexity. – Fedor Nikitin Apr 18 '14 at 14:40