  Self-concordance is the most important property required for barriers in
convex programming. It is intrinsically linked to the affine structure of the
underlying space. Here we introduce an alternative notion of self-concordance
which is linked to the projective structure. A function on a set $X \subset
A^n$ in an $n$-dimensional affine space is projectively self-concordant if and
only if it can be extended to an affinely self-concordant logarithmically
homogeneous function on the conic extension $K \subset V^{n+1}$ of $X$ in the
$(n+1)$-dimensional vector space obtained by homogenization of $A^n$. The
feasible sets in conic programs, notably linear and semi-definite programs, are
naturally equipped with projectively self-concordant barriers. However, the
interior-point methods used to solve these programs employ only affine
self-concordance. We show that estimates used in the analysis of interior-point
methods are tighter for projective self-concordance, in particular inner and
outer approximations of the set. This opens the way to a better tuning of
parameters in interior-points algorithms to allow larger steps and hence faster
convergence. Projective self-concordance is also a useful tool in the
theoretical analysis of logarithmically homogeneous barriers on cones.
