Convex expansion of condenser plates

Abstract

In this paper we introduce an ew notion of the con vex expansion (convex hull) of condenser plates invariant with respect to M ¨ obius transformations of the space. If one plate is the point at infinity, then the convex expansion of another bounded plate coincides with its usual convex hull. The convex expansion of plates does not decrease the condenser capacity. The main theorem shows that in the case of a flat condenser with connected plates it cannot increase more than threefold. In particular, if the monitoring of the hidden deformation of a condenser with flat plates (with the constancy of its M ¨ obius invariant hull) shows the decrease of its capacity more than twice, then this indicates to the violation of the connectivity, i.e., the destruction of the plate. This result is applicable to nondestructive testing.