Brunn-Minkowski Inequality

Definition 1
 

Let \( \mathcal U \) and \( \mathcal V \) be two sets of points in the plane. We will define their half-sum \( \mathcal U\oplus \mathcal V \) as:

\( \mathcal U\oplus \mathcal V=\{X: X \) is the midpoint of the segment \( AB \) where \( A \) and \( B \) are arbitrary points of \( \mathcal U \) and \( \mathcal V \) respectively.\( \} \)

Problem 1
 

Assume that \( \mathcal U=\{U\} \) is a single point and \( \mathcal V \) is an arbitrary set. Then \( \mathcal U\oplus \mathcal V \) is the image of \( \mathcal V \) under under the homothety with center \( U \) and coefficient \( \frac12 \).

To any set of points \( \mathcal U \) in the plane we can be associate the following set of vectors \( \overrightarrow{\mathcal U}=\{\overrightarrow{OX}: X\in \mathcal U\} \), where \( O \) is a fixed point in the plane (in the sequel we will call it the origin). From now on let us assume that \( O \) is some fixed point in the plane. For any two sets \( \mathcal U \) and \( \mathcal V \) in the plane we define the sum of the sets \( \overrightarrow{\mathcal U} \) and \( \overrightarrow{\mathcal V} \) as: \[ \overrightarrow{\mathcal U}+\overrightarrow{\mathcal V}=\{ \overrightarrow u+\overrightarrow v: \overrightarrow u\in \overrightarrow{\mathcal U}, \overrightarrow v\in \overrightarrow{\mathcal V}\}.\]

Problem 2
 

Let \( \mathcal U=\{U\} \) be a single point, and \( \mathcal V \) an arbitrary set in the plane. If we consider all vectors to originate from \( O \), then the set \( \overrightarrow{\mathcal U}+\overrightarrow{\mathcal V} \) is a translation of \( \overrightarrow{\mathcal V} \) for the vector \( \overrightarrow{OU} \).

Theorem 1
 

The set of all endpoints of the vectors from \( \overrightarrow{\mathcal U}+ \overrightarrow {\mathcal V} \) (if those vectors are set to originate from \( O \)) is similar to the set \( \mathcal U\oplus \mathcal V \) with the coefficient of similarity \( \frac12 \).

By its definition \( \overrightarrow{\mathcal U}+ \overrightarrow {\mathcal V} \) is a set of vectors. If we consider these vectors as vectors originating from \( O \), we can look at the set of their endpoints. Let us denote this set by \( \mathcal U+ \mathcal V \). We saw that this set is similar to \( \mathcal U\oplus \mathcal V \) and twice as large. Therefore many of the theorems about \( \mathcal U+ \mathcal V \) will be identical (or very analogous) to the theorems about \( \mathcal U\oplus \mathcal V \). Let us also notice that if one of the sets \( \mathcal U \) and \( \mathcal V \) is translated, then their sum and the half-sum will also be translated by the appropriate vector.

Remark. The sum \( \mathcal U+\mathcal V \) is often called the Minkowski sum of the sets \( \mathcal U \) and \( \mathcal V \).

Problem 3
 

Assume that \( \mathcal U \) and \( \mathcal V \) are convex sets. Prove that \( \mathcal U\oplus \mathcal V \) and \( \mathcal U+ \mathcal V \) are convex as well.

The following problems are not difficult, but it is important for you to do them. Solving them will make you more familiar with the Minkowski sum and you will have an easier time reading the rest of the paper.

Problem 4
 

Find \( \mathcal U\oplus \mathcal V \), if:

  • (a) \( \mathcal U \) and \( \mathcal V \) are parallel segments \( AB \) and \( CD \).

  • (b) \( \mathcal U \) and \( \mathcal V \) are non-parallel segments \( AB \) and \( CD \).

  • (c) \( \mathcal U \) and \( \mathcal V \) are two rectangles \( ABCD \) and \( PQRS \) such that \( AB\|PQ \).

Problem 5
 

Find examples of pentagons and triangles whose sums are pentagon, \( 6 \)-gon, \( 7 \)-gon, and \( 8 \)-gon.

Theorem 2 (Brunn-Minkowski)
 

If \( \mathcal U \) and \( \mathcal V \) are convex sets in the plane, then \[ \sqrt{S_{\mathcal U+\mathcal V}}\geq \sqrt{S_{\mathcal U}}+\sqrt{S_{\mathcal V}}.\]

Problem 6 (Dušan Djukić, IMO 2006)
 

Assign to each side \( b \) of a convex polygon \( \mathcal P \) the maximum area of a triangle that has \( b \) as a side and is contained in \( \mathcal P \). Show that the sum of the areas assigned to the sides of \( \mathcal P \) is at least twice the area of \( \mathcal P \).


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