Title: Self-dual polyhedral cones in R^4.

A cone is self-dual if it equals its dual. For example an orthant is self-dual. A less trivial example is the cone of positive semidefinite matrices. Recently João Gouveia and Bruno F. Lourenço studied self-dual polyhedral cones and showed that in R^4 the 2-dimensional faces of a self-dual cone must correspond to a 3-connected planar graph which is self-dual in the plane and furthermore this self-duality is strongly involutive. They conjectured that the reverse also holds, that is, for each 3-connected strongly involutive self-dual graph there is a self-dual cone realizing it. We show that the conjecture is true. We also discuss how this result corresponds to characterizations of diameter graphs, touching graphs of circles, ball-polytopes, quadrangulations of the projective plane and generalized thrackles.