Final answer:
The polar cone K° is the set of all vectors that satisfy the condition that the inner product of a vector in K° and every vector in K is less than or equal to zero. It can be expressed as the intersection of half-spaces defined by the inequalities in K. For example, in the case of n=2, K° is the reflection of K about the origin in the two-dimensional plane.
Step-by-step explanation:
The polar cone of set K is denoted as K°. The polar cone is the set of all vectors that satisfy a specific condition with respect to each vector in K.
In this case, the condition is that the inner product of a vector in K° and every vector in K is less than or equal to zero. Since K is defined as a set of vectors where each component is non-decreasing, we can define the polar cone K° as:
- A vector v is in K° if and only if for every vector x in K, the inner product of v and x is less than or equal to zero.
- Since each component of K is non-decreasing, we can express K as a set of half-spaces, where each half-space corresponds to a specific component inequality. Then, the polar cone K° can be expressed as the intersection of all half-spaces defined by the inequalities in K.
- For example, let's consider n = 2. In this case, K can be represented as the set of points in the first quadrant of the x₁-x₂ plane. The polar cone K° will be the set of points in the third and fourth quadrants of the plane, which is the reflection of K about the origin.