If I collapse a (say, closed) set in a length space, I obtain a length space: is there some literature on this?

We consider length spaces as defined by Gromov and others. [However the case of a Riemannian distance already leads to interesting examples of what I am writing]. If $X$ is one such space, it has a distance $d$ which can be recovered by the length of curves. Suppose $X$ is path connected. Let $c:[a,b]\to X$ be a curve, define $|c'(t)|=\lim_{\epsilon\to0}\sup_{|u-t|,|v-t|\le\epsilon}\frac{d(c(u),c(v))}{|u-v|}$ and $$ L(c)=\int_a^b|c'(t)|dt, $$ and set $$ D(x,y)=\inf\{L(c) \text{ $c$ is a curve having endpoints $x$ and $y$}\}. $$ We are interested in cases where $d=D$ and where the distance $D=d$ is realized by lengths of geodesics (i.e. the hard work has been already done).

Now, let $E\subseteq X$ be closed and define a distance $D_X$ on $X\setminus E\cup\{E\}$ ($X$ with $E$ collapsed to a point): $$ L_E(c)=\int_a^b|c'(t)|\chi_{X\setminus E}(c(t))dt, $$ and set $$ D_E(x,y)=\inf\{L_E(c) \text{ $c$ is a curve having endpoints $x$ and $y$}\}, $$ if $x,y\in X\setminus E$, and $D_E(x,E)=D(x,E)=\inf_{y\in E}D(x,y)$.

The space $X\setminus E\cup\{E\}$ is clearly a length space w.r.t. the distance $D_E$.

The reason I find these objects interesting is that, if $E$ is the smooth boundary of an open subset of $X$, a nice Riemannian manifold, then the points of $E$ play the role of the unit vectors in the tangent space of a point; this meaning that they can parametrize those geodesics leaving $E$ which, at least locally, minimize the distance from $E$. Even in the case of the Euclidean plane one obtains interesting pictures.

I would be very surprised if no one had developed this viewpoint in the past.

up to reparametrizations, so you would have $D(x,y)=0$ for all $x,y$! However, the equality "integral of |c'(t)| = length" is true for Lipschitz curves, which is ok for many purposes since continuous curves, finite length curves are Lipschitzup to reparametrization. $\endgroup$ – Pietro Majer Jun 12 at 7:42