Quermass-interaction processes: Conditions for stability

We consider a class of random point and germ-grain processes, obtained usin g a rather natural weighting procedure. Given a Poisson point process, on e ach point one places a grain, a (possibly random) compact convex set. Let X i be the union of all grains. One can now construct new processes whose den sity is derived from an exponential of a linear combination of quermass fun ctionals of Xi. If only the area functional is used, then the area-interact ion point process is recovered. New point processes arise if we include the perimeter length functional, or the Euler functional (number of components minus number of holes). The main question addressed by the paper is that o f when the resulting point process is well-defined: geometric arguments are used to establish conditions for the point process to be stable in the sen se of Ruelle.