  We explore a generalisation of the L\'evy fractional Brownian field on the
Euclidean space based on replacing the Euclidean norm with another norm. A
characterisation result for admissible norms yields a complete description of
all self-similar Gaussian random fields with stationary increments. Several
integral representations of the introduced random fields are derived. In a
similar vein, several non-Euclidean variants of the fractional Poisson field
are introduced and it is shown that they share the covariance structure with
the fractional Brownian field and converge to it. The shape parameters of the
Poisson and Brownian variants are related by convex geometry transforms, namely
the radial $p$th mean body and the polar projection transforms.
