  This paper studies extended formulations for radial cones at vertices of
polyhedra, where the radial cone of a polyhedron $ P $ at a vertex $ v \in P $
is the polyhedron defined by the constraints of $ P $ that are active at $ v $.
Given an extended formulation for $ P $, it is easy to obtain an extended
formulation of comparable size for each its radial cones. On the contrary, it
is possible that radial cones of $ P $ admit much smaller extended formulations
than $ P $ itself. A prominent example of this type is the perfect-matching
polytope, which cannot be described by subexponential-size extended
formulations (Rothvo\ss{} 2014). However, Ventura & Eisenbrand (2003) showed
that its radial cones can be described by polynomial-size extended
formulations. Moreover, they generalized their construction to $ V $-join
polyhedra. In the same paper, the authors asked whether the same holds for the
odd-cut polyhedron, the blocker of the $ V $-join polyhedron. We answer this
question negatively. Precisely, we show that radial cones of odd-cut polyhedra
cannot be described by subexponential-size extended formulations. To obtain our
result, for a polyhedron $ P $ of blocking type, we establish a general
relationship between its radial cones and certain faces of the blocker of $ P
$.
