We will discuss the following topics, which happen to be closely related:
1. What are the restrictions on the Gaussian measures of closed surfaces (or surfaces with planar boundaries). For instance, under what conditionscan one construct a polyhedron (for instance, a two-dimensionalpolyhedral surface in Rr) with given areas and directionsof faces and no boundary (or a planar boundary).
2. What is the relationship between different types of ellipticity for surfacearea functionals (an area functional is elliptic over Z (resp. R) if regions in affine planes are area minimizers among all Lipschitz chains over Z (R) with the same boundary).
3. Optimal fillings: metrics on a manifold (with boundary) that admit novolume-decreasing perturbations that do not decrease distances betweenboundary points.
4. Asymptotic growth of volume for large balls in a periodic metric ( a metricinvariant under a co-compact action of an abelian group).
The talk is based on a joint work with S. Ivanov.