Let $X$ be a subset of some Euclidean space. We say that $X$ is convex if for any two points $p$, $q$ in $X$, the line segment joining $p$ and $q$ is also in $X$. But what if we loosen this definition a bit, and say that $X$ is $n$-convex if for every $p$, $q$, there is a piecewise-linear path from $p$ to $q$ consisting of at most $n$ segments? In this way $1$-convexity would just be convexity. Star convexity would (strictly) be a bit stronger than $2$-convexity. This would be something that's $3$-convex:
An infinite spiral shape wouldn't be $n$-convex for any $n$.
Have people studied this notion of "$n$-convexity"? This isn't homework or research or anything, I was just curious and I didn't find a quick answer on wikipedia or google.