The definition of convexity for a body in $\mathbb{R}^n$ is simple enough, namely: a body $K\subset \mathbb{R}^n$ is convex if for any two points $x,y\in K$ the line segment between $x$ and $y$ is entirely contained in $K$.
However supposing $K$ is not convex, is there a standard definition of it's "level of convexity", i.e. answering the question "how close is $K$ to being convex"? Would such a definition even be useful? If the answer to both questions is "Yes", what is it used for and is there some relevant literature? If not, why not?
Here are some fairly elementary ideas I've had so far. The proposed "level of convexity" in suggestion $i$ is denoted as $C_i(K)$. To avoid complications I assume $K$ is connected. I should note that none of these are trivial to compute (at least not for me).
1) Combinatorial approach (if not one line, how many lines?): for any $x,y\in K$ define $d_K(x,y)$ as the smallest number of adjacent line segments contained in $K$ needed to walk from $x$ to $y$ (this is in fact a metric - albeit a rather clumsy one). Then $C_1(K)=\underset{x,y\in K}{\max}d_K(x,y)$. For example: for a convex set $C_1(K)=1$, for non-convex star domains $C_1(K)=2$.
2) The geometric approach (how much extra volume do we need?): let $M$ be the convex hull of $K$ in $\mathbb{R}^n$, then $C_2(K)=\frac{vol_n(K)}{vol_n(M)}$.
3) Probabilistic approach (how likely is it that we could draw the line inside K?): let $Z:K\times K \rightarrow \{0,1\}$ be a random variable taking $Z(x,y)=1$ iff the line segment between $x$ and $y$ is contained in $K$. Then $C_3(K)=\mathbb{E}[Z]$.
4) Probabilistic-Geometric approach (how much of the line is likely to stay inside?): for any $x,y\in K$ denote $l_{xy}$ as the line segment between $x$ and $y$. let $Z:K\times K \rightarrow \{0,1\}$ be a random variable taking $Z(x,y)=\frac{|l_{xy}\cap K|}{|l_{xy}|}$. Then $C_4(K)=\mathbb{E}[Z]$.
Images: Image 1 is a trivial example of a shape for which $C_2(K)\approx 1$ but $C_3(K)\approx \frac{1}{2}$. For $C_2$ and $C_4$ the fact that they're not equivalent isn't so immediate, but my calculations for the body in Image 2 yielded $C_2(K)=\frac{5}{7}$ and $C_4(K)\approx 0.891$, though it's entirely possibly my calculations were incorrect (and even if they are one could still conjecture that $C_2$ and $C_4$ are dependent in some strong sense).
A measure on non-convexity that is used in the theory of convex bodies is the following: if $A\subset\mathbb{R}^n$, define $$ c(A) := \inf\{\lambda \geq 0:\ A + \lambda\, \text{conv} \, A\ \text{is convex}\}. $$ By the Shapley-Folkman theorem, one has $0\leq c(A) \leq n$. Moreover, if $A$ is compact, then $c(A) = 0$ if and only if $A$ is convex, whereas $c(A) = n$ if and only if $A$ consists of $n+1$ affinely independent points.
See the book of Schneider, "Convex bodies...", p. 131, for details.