Assume that S CR is a closed convex set. Let x = R and y =...

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Assume that S CR is a closed convex set. Let x = R and y = Rn be two points with x & S and y & S. Let x = S and y = S be the unique minimum distance points (i.e., projections) of x and y, respectively, in the set S. Show that ||T Y|| ||x y|| (1) Hint: Use the Closest-point Theorem presented in Lecture 2. You may also want to draw a picture by hand to clearly understand the meaning of (1). We start with how to identify the closest points to sets. Theorem 5.1. Closest-point theorem Let S be a closed convex set in R^ and y & S. Then, there exists a unique point S with minimum distance from y. In addition, is the minimising point if and only if (y-T) (x) < 0, for all x S Simply put, if S is a closed convex set, then S will be the closest point to y & S if the vector y - T is such that if it forms an angle that is greater or equal than 90 with all other vectors - I for x = S. Figures 8 illustrates this logic. Y S X S T Figure 8: Closest-point theorem for a closed convex set (on the left). On the right is an illustration of how the absence of convexity invalidates the result. - Notice that S lies in the half-space (y )(x ) 0 defined by the hyperplane p (x ) = 0 with normal vector p = (y). We will next revise the concepts of half-spaces and hyperplanes since they will play a central role in the derivations in this course. Assume that S CR is a closed convex set. Let x = R and y = Rn be two points with x & S and y & S. Let x = S and y = S be the unique minimum distance points (i.e., projections) of x and y, respectively, in the set S. Show that ||T Y|| ||x y|| (1) Hint: Use the Closest-point Theorem presented in Lecture 2. You may also want to draw a picture by hand to clearly understand the meaning of (1). We start with how to identify the closest points to sets. Theorem 5.1. Closest-point theorem Let S be a closed convex set in R^ and y & S. Then, there exists a unique point S with minimum distance from y. In addition, is the minimising point if and only if (y-T) (x) < 0, for all x S Simply put, if S is a closed convex set, then S will be the closest point to y & S if the vector y - T is such that if it forms an angle that is greater or equal than 90 with all other vectors - I for x = S. Figures 8 illustrates this logic. Y S X S T Figure 8: Closest-point theorem for a closed convex set (on the left). On the right is an illustration of how the absence of convexity invalidates the result. - Notice that S lies in the half-space (y )(x ) 0 defined by the hyperplane p (x ) = 0 with normal vector p = (y). We will next revise the concepts of half-spaces and hyperplanes since they will play a central role in the derivations in this course.