  Not all convex functions on $\mathbb{R}^n$ have finite minimizers; some can
only be minimized by a sequence as it heads to infinity. In this work, we aim
to develop a theory for understanding such minimizers at infinity. We study
astral space, a compact extension of $\mathbb{R}^n$ to which such points at
infinity have been added. Astral space is constructed to be as small as
possible while still ensuring that all linear functions can be continuously
extended to the new space. Although astral space includes all of
$\mathbb{R}^n$, it is not a vector space, nor even a metric space. However, it
is sufficiently well-structured to allow useful and meaningful extensions of
concepts of convexity, conjugacy, and subdifferentials. We develop these
concepts and analyze various properties of convex functions on astral space,
including the detailed structure of their minimizers, exact characterizations
of continuity, and convergence of descent algorithms.
