Convex Set: A set of points K in the xy-plane is called convex if for each pair of points A,B in K, the intervening line segment AB is also contained in K. Closed Convex Curve: If the convex set K is bounded and has non-empty interior points, the boundary of K is called a closed convex curve. The boundary of K is denoted by . The length of is called the perimeter of K.

Note that if a planar set is convex and yet has no boundary points, then it must be the entire plane. There are of course unbounded planar convex sets which do not equal the entire plane: can you think of one?

Examples

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Non-Examples

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Minkowski Sum (Mixed Area of Minkowski): Suppose A and B are bounded convex sets in the xy-plane. The Minkowski Sum of sets A and B is the set .

Problem 5 (0) Let A and B be two convex sets. Prove that the Minkowski Sum of A and B is convex.

Example: Suppose D is a convex set in the xy-plane, and B(0,r) is the ball centered at the origin with radius r, as shown in the figure below.

Thus, the Minkowski Sum of D and B(0,r) is , which is graphed below.

Parallel Convex Sets: The outer parallel set Kr of a convex set K is the union of all the closed balls of radius r, centered at the points of K. The boundary is called the outer parallel curve of in the distance r.

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Support Lines: Suppose K is a convex set in the xy-plane, and P is an element of the boundary. A line L though P is called a support line of K if K lies, completely, in one of the two regions into which the line L divides the plane.

Support lines are very similar to the concept of tangent lines that were introduced in calculus. There is a subtle difference, however, in that support lines exist at corner points where tangent lines do not, as they depend on the value of the derivative at that point, which of course does not exist. Moreover, a corner point is defined in this course as a point of where there are more than one support line at that point. A support line for any non-corner point in a convex set is it's tangent line.

Let K be a bounded convex set in the xy-plane, let OR be a ray directed from the origin O, and let be the angle of OR with repect to the x-axis. Suppose that G₁(p₁,) is any line that intersects K and is perpendicular to OR (e.g., the dotted line in the figure below). p₁ is the distance from the origin to the line G₁(p₁,). Let p be the maximum value of the set . In other words, we are searching for a line farthest from the origin that is perpendicular to OR at angle and intersects K at only one point. Then the corresponding line, G(p,) is the support line of K in the direction of . By construction, the function is a continuous function of as ranges from 0 to 2. The function is called the support function of the convex set K.

We want to find an equation for G. First let's review the dot product and orthogonal projections. Let = (a,b) and let = (c,d). Then .

Given a vector, , and the unit vector, , the orthogonal projection of onto is defined by the equation: , where is the length of the orthogonal projection.

Problem 5 (1) Let = (3,4) and let = (2,4). What is the orthogonal projection of onto ?

We want find all the vectors that project whose projection onto the unit vector has length p. This infinite collection of vectors, i.e. = p will then give us the line G. The unit vector is given by =, indicated in red. We have . The equation for G is thus and is called the generalized normal equation for G.

What about the support line at angle closest to the origin? This support line is actually for the angle . Note that the line OR at angle has the same slope as a line drawn at angle , but the positive orientation is pointing in the opposite direction. In this particular example, when is the angle, the positive distance, p is to the right of the origin, whereas at , the positive distance p is to the left of the origin.

Breadth is a concept you are sure to be familiar with; it's the measure of the "distance" between an object's sides. When in reference to convex sets in the xy-plane, breadth is defined as the distance between two parallel support lines. As shown above, if one support line is said to be in the direction , then its parallel counterpart is in the direction .

Support Line along the direction Suppot Line along the direction

Clearly the breadth depends on the the angle, as illustrated in the picture below. Each set of colored parallel lines is a different value of and its parallel counterpart. The width of a convex set is defined as the shortest breadth, and is given by the equation: . (Recall that the distance from the origin to the support line along the direction actually has a negative value; thus the change in sign). The function is called the width function of the convex set K. (Note this is a smooth curve with no corner points). The largest breadth of a convex set K is known as the diameter of a convex set. If constant for all , the convex set K is said to be of constant breadth.

The circle is a great example of a convex set with constant breadth. Suppose our circle centered at (a,b) and has radius r. The implicit equation for the circle is thus . If we change to polar coordinates we have and , or parametric equations for the circle. It follows that

Other than the circle, the simpler convex sets of constant breadth are known as the Reuleaux polygons. Below is an example of the Reuleaux triangle, where each vertex is the center of a circle whose radius is equal to the length of a side of the triangle.

Fact #1: Every convex set in the xy-plane has a boundary that is piecewise differentiable.

Piecewise Differentiable: a real-valued function f : R->Rn is piecewise differentiable on a closed interval I=[a,b] provided there are a finite number of points ck, where c0=a and cn=b, such that f is differentiable for all x that satisfy ck -1 < x < ck and k=1...n.

Fact #2: Every point on the boundary of a convex planar region (that is not the entire plane) has at least one support line.

Problem 5 (2) Every (finite or infinite) intersection of convex sets is convex (3) is convex if are all convex and (4) If and are the support functions for the convex sets K and L, respectively, what is the support function for K+L. (5) Using the support function, prove that a circle centered at (2,2) with radius 1 has a width of 2. Solutions

References: De-lin, Santalo.