C] is the mean chord length of a convex region of area A bounded by a curve L of length L, and [bar.
C] of a convex region of volume V, bounded by a closed convex surface S of surface area S:
2] - [pi]A, where [alpha] denotes the angle between two tangents from an exterior point (x,y) to the curve of the length L that forms the boundary of a convex region of the area A, and the integral is taken over the whole plane outside the boundary.
A convex region
covers a family of arcs if it contains a congruent copy of every arc in the family.
Consequently, the simplified line cannot stray from the convex region determined by vertices of the original line.
For point-line conflicts we must examine points that lie in certain restricted convex regions; for line-line conflicts we need only examine segment end-points that lie in certain restricted convex regions.
This corollary permits a search strategy that (1) conveniently limits the search space to nested sets of convex regions, and (2) examines only line end-points, not the lines themselves.
It is clear from the figure that commodity units with quality index values in the convex region may benefit from some purification, i.
Equation (1) says that in the convex region the market provides an incentive to separate the good from the bad because the total value of the parts (i.
Duran's topics include general properties of harmonic mappings, harmonic mappings into convex regions
, harmonic self-mappings of the disk, harmonic univalent functions, external problems, mapping problems, minimal surfaces and curvature of minimal surfaces, with particular attention to the Weierstrass-Enneper representation.