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_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.