Thm 3.1: The intersection of two convex sets is a convex set.
A plane curve is the image of a mapping from the unit interval to the plane which is continuous in both coordinates. If the mapping is one-to-one, the curve is simple. If the endpoints coincide the curve is closed. We wish to discuss the inside and outside of the plane region determined by a simple closed curve. To do this precisely we need some more concepts.
Def: The open circular neighborhood of a point P with radius r, N(P,r). Also closed neighborhood.
Interior point of a set.
Exterior point of a set.
Boundary point of a set.
An open set has only interior points.
A closed set contains all of its boundary points.
Emphasize that these are not mutually exclusive concepts.
Bounded set.
Def: A supporting line for a two dimensional set with interior points is a line through at least one boundary point of the set such that all points of the set are in the same closed half-plane determined by the line.
Thm. 3.3: A line is a supporting line for a convex set if it goes through at least one boundary point but no interior point of the set, and conversely.
Def: The tangent cone for a boundary point of a convex set is the set of all rays that:
Def: A line is the tangent to a convex set at a point if it is the union of two collinear semi- tangents at that point.
Note that this concept of tangent is different from the one used in calculus.
Boundary points of a convex set can be classified as regular or corner,
Thm. 3.4 : The interior of a simple closed curve is a convex set if and only if through each point of the curve there passes at least one supporting line for the interior.
Supporting half-plane.
Thm. 3.5 : A plane closed convex set that is a proper subset of a plane is the intersection of all its supporting half-planes.
Def: A convex body is a convex set of points that is closed, bounded, nonempty and has interior points.
Thm. 3.6 : A simple closed curve S and its interior form a convex body K if and only if every line through an interior point of K intersects S in exactly two points.
Thm 3.7 : The boundary of a convex body in two-space is a simple closed curve. (No proof)
Thm 3.8 : A simple closed curve S is the boundary of a 2-dim convex body K if and only if each closed closed polygon determined by successive points on S is the boundary of a convex polygonal region.
Def: The length of a simple closed curve is the least upper bound of the length of all inscribed polygonal curves. The length of a boundary of a region is called the perimeter of the region.
Thm 3.9 : If K1 and K2 are convex polygonal regions with K1 {intersect} K2, then the perimeter of K1 is less than or equal to the perimeter of K2.
Thm 3.10: Same for 2-dim convex bodies.
Thm 3.11: If a function ax + by + c is defined for each point of a convex polygonal region, the maximum value occurs for the coordinates of one vertex and the minimum value occurs for the coordinates of another vertex.
Linear programming
Def: The convex hull of a set S is the smallest convex set containing S.
Thm 3.16: A set is convex if and only if it is its own convex hull.
Def: A point A of convex set K is called an extreme point of K iff it is not an interior point of any line segment contained in K.
Thm 3.17 : A convex polygonal region S is the convex hull K of its extreme points.
Thm 3.18 : The convex hull of a finite number of points in the plane is a convex polygonal region.
Thm 3.19: Let S be a finite set of points on a simple closed curve T. T is the boundary of a convex set K iff for all such sets S, no point of S is an interior point of the convex hull of K.
Def: The perpendicular distance along a line l between two parallel supporting lines of a bounded 2-dim set is the width of the set in the direction indicated by line l.
Thm 3.21: Let {pi} and {pi}' be parallel supporting planes for a set S in a direction of maximum width. If A is any point of {pi intersect} S, then the line l through A perpendicular to {pi} and {pi}' intersects {pi}' in a point B of S.
Thm 3.22: Let {pi} and {pi}' be parallel supporting planes for a closed set S in a direction of maximum width. Then {pi intersect} S and {pi' intersect} S each contain exactly one point.
Def: If the width of a set is the same in all directions, we say that the set is a set of constant width.
Convex sets of constant width are of practical interest. Disks are an example, but the most interesting example is the Reuleaux triangle.
Construction of Reuleaux triangle.
Applications : Wankel engine, used in some autos and in snowmobiles. Gear in driving a movie film. Drill that makes square holes. "Jimmy-proof" fire hydrants.
Other sets of constant width.