  We investigate spacetimes whose light cones could be anisotropic. We prove
the equivalence of the structures: (a) Lorentz-Finsler manifold for which the
mean Cartan torsion vanishes, (b) Lorentz-Finsler manifold for which the
indicatrix (observer space) at each point is a convex hyperbolic affine sphere
centered on the zero section, and (c) pair given by a spacetime volume and a
sharp convex cone distribution. The equivalence suggests to describe {\em
(affine sphere) spacetimes} with this structure, so that no algebraic-metrical
concept enters the definition. As a result, this work shows how the metric
features of spacetime emerge from elementary concepts such as measure and
order. Non-relativistic spacetimes are obtained replacing proper spheres with
improper spheres, so the distinction does not call for group theoretical
elements. In physical terms, in affine sphere spacetimes the light cone
distribution and the spacetime measure determine the motion of massive and
massless particles (hence the dispersion relation). Furthermore, it is shown
that, more generally, for Lorentz-Finsler theories non-differentiable at the
cone, the lightlike geodesics and the transport of the particle momentum over
them are well defined though the curve parametrization could be undefined.
Causality theory is also well behaved. Several results for affine sphere
spacetimes are presented. Some results in Finsler geometry, for instance in the
characterization of Randers spaces, are also included.
