Simple compactifications and polar decomposition of homogeneous real spherical spaces

Let Z be an algebraic homogeneous space Z=G/H attached to real reductive Lie group G. We assume that Z is real spherical, i.e., minimal parabolic subgroups have open orbits on Z. For such spaces we investigate their large scale geometry and provide a polar decomposition. This is obtained from the existence of simple compactifications of Z which is established in this paper.
Comment: Extended revised version. Section 5 with a systematic study of the compression (or valuation) cone is new