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Two dimensional steepest descent curves (SDC) for a quasi convex family areconsidered; the problem of their extensions (with constraints) outside of aconvex body $K$ is studied. It is shown that possible extensions areconstrained to lie inside of suitable bounding regions depending on $K$. Theseregions are bounded by arcs of involutes of the boundary of $K$ and satisfymany inclusions properties. The involutes of the boundary of an arbitrary planeconvex body are defined and written by their support function. Extensions SDCof minimal length are constructed. Self contracting sets (with oppositeorientation) are considered, necessary and/or sufficients conditions for themto be subsets of a SDC are proved.

Bounding regions to plane steepest descent curves of quasi convex families