About Minkowski-inequality

Maximalization problem

Problem 1

Prove the following inequality for the a, b, e, f positive numbers. When will equality hold?

Solution:

To square it two times and we will get the following one:

According to the conditions the two inequalities are equivalent, the latter one is obviously true. Equality holds if and only if

Problem 2

Prove the following inequality for the a, b, c, e, f, g positive numbers. When will equality hold?

Solution:

The above method is not passable in this case. We will show a very different type of solution.

Take into the point A two mutually perpendicular lines. Place from the point A on one line the section with length a, next the section with length b and the section with length c, as well as on the other line the section with legth e, the section with length f, the section with length g, see figure below. The red lines starts from the end the sections and are parallel to the other perpendicular line. The intersections of the red lines are points: A, B, C, D.

By Pythagorean theorem the left side of the inequality is the length of the ABCD Polyline and the right side of the inequality is the length of AD section.

The triangle-inequality on the basis we are ready!

Equality holds if and only if the Polyline sections are parallel to the section AD, that is all of this sections are parallel to each other, so if:

Note 1

We can prove the following inequality using the same method. We have to increase the number of the sections to n:

where equality holds if and only if :

Note 2

There are the special cases of the Minkowski inequality.