Minkowskis theorem is a theorem describing the relationship between a two-dimensional convex body and its centroid. The theorem was discovered by Hermann Minkowski in 1901. In its simplest form, the theorem states that for any two-dimensional convex body, there is a unique point inside the body, referred to as the centroid, from which the distances of every point to all other points is equal or minimal.
In its general form, Minkowskis theorem states that:
For any convex body in an n-dimensional Euclidean space, there is a point inside the body, referred to as the centroid, such that the sum of the distances from this point to all other points in the body is minimal.
Proof
Consider a two-dimensional convex body, B. Let A be the set of all points inside B. Every point P in A has a unique distance d from every other point in A. To prove that the centroid exists, it is sufficient to prove that there is a point P0 in A, such that d is minimized when P0 is the centroid.
Let P1, P2, P3, ..., PN be the points in A. Set the distance d between P1 and all other points in A to be d1. For each d1, there are two possible states - the point P0 is the centroid, or it is not. We will first assume that the point P0 is the centroid, and then prove that it is the best case.
Given that P0 is the centroid, the distance between P1 and all other points in A must be equal and equal to d1. Thus, the sum of the distances from P1 to all other points in A is N x d1. Next, we assume that the point P0 is not the centroid, and consider the total distance from P1 to all other points.
If P0 is not the centroid, then the distances from P1 to all other points in A might not be the same. Specifically, let d2, d3, ..., dN be the distances from P1 to all other points in A, such that d2 > d1, d3 > d1, and so on. Thus, the sum of the distances from P1 to all other points in A is
d1 + d2 + d3 + ... + dN
The total distance is greater than N x d1 when P0 is not the centroid, thus it follows that the centroid must minimize the total distance between each point in A.
To prove the theorem in general, the proof is similar. Every point in an n-dimensional space has a unique distance d from every other point. Let the distances between P1 and all other points in A to be d1. The total distance from P1 to all other points can be minimized when P0 is the centroid, as demonstrated in the proof for two-dimensional convex bodies. Thus, the centroid must minimize the total distance from every point in A.
Conclusion
Minkowskis theorem is an important theorem in convex geometry. It states that for any convex body in an n-dimensional Euclidean space, there is a unique point inside the body, referred to as the centroid, from which the sum of the distances to all other points is minimal. The theorem provides a method for computing the centroid of a convex body and is used in many applications, such as solving optimization problems.
