Question

Let A be a compact subset of Rd. For x ∈ Rd a fixed point'e, show that there exists a

a ∈ A such that minimizes the distance to x, i.e. such that
∀y ∈ A, ∥x - a∥ ⩽ ∥x - y∥
The value of this minimum, ∥x - a∥ is called the distance between point x and compact A.

Answer 1

For a fixed point x and a compact subset A in Rd, there exists a point a in A that minimizes the distance to x. This minimum distance is called the distance between point x and compact A.

Step-by-step explanation:

In mathematics, for a fixed point x in Rd and a compact subset A of Rd, it can be shown that there exists a point a in A that minimizes the distance to x, denoted as ||x - a||. This means that for every point y in A, the distance between x and a is less than or equal to the distance between x and y.

To prove this, we can use the fact that A is a compact subset of Rd. Since A is compact, it is also closed and bounded. We can then construct a sequence of points in A that converges to x. Since A is closed, the limit point of this sequence must also be in A. The point in A that is closest to x is the limit point of this sequence.

Therefore, we can conclude that for any fixed point x and compact subset A in Rd, there exists a point a in A that minimizes the distance to x, and this minimum distance is called the distance between point x and compact A.