A smooth enough function \(f: D \rightarrow {\mathbb {R}}\) defined on a domain \(D \subset V\) in a real vector space is an analytic object which defines two different geometric objects. On the one hand, one may consider its level hypersurfaces as affine hypersurface immersions in V, on the other hand the graph of its gradient \(\nabla f\) as a Lagrangian immersion in the product \(V \times V^*\) of the real space with its dual. We study these geometric interpretations for the class of self-concordant logarithmically homogeneous barriers defined on the interior of regular convex cones. Properties of the barrier such as convexity, self-concordance, and self-scaledness are equivalent to meaningful properties of the geometric objects. This equivalence furnishes new vantage points to study barriers in conic optimization and builds a bridge to other areas of mathematics which open new ways to obtain results in optimization. We describe the links and equivalences between the three different view-points, the analytic one and the two geometric ones, and give examples of results obtained by means of these connections.
