Bounding Regions to Plane Steepest Descent Curves of Quasiconvex Families

Abstract

Two-dimensional steepest descent curves (SDC) for a quasiconvex family are considered; the problem of their extensions (with constraints) outside of a convex body $K$ is studied. It is shown that possible extensions are constrained to lie inside of suitable bounding regions depending on $K$. These regions are bounded by arcs of involutes of $\partial K$ and satisfy many inclusions properties. The involutes of the boundary of an arbitrary plane convex body are defined and written by their support function. Extensions SDC of minimal length are constructed. Self-contracting sets (with opposite orientation) are considered: necessary and/or sufficient conditions for them to be subsets of SDC are proved.