The original version of the Brunn–Minkowski theorem (Hermann Brunn 1887; Hermann Minkowski 1896) applied to convex sets; the generalization to compact nonconvex sets stated here is due to Lazar Lyusternik (1935).
Then the following inequality holds: where A + B denotes the Minkowski sum: The theorem is also true in the setting where
See the discussion in Gardner's survey for more on this, as well as ways to avoid measurability hypothesis.
This condition is not part of the multiplicative versions of BM stated below.
We give a well-known argument that follows a general recipe of arguments in measure theory; namely, it establishes a simple case by direct analysis, uses induction to establish a finitary extension of that special case, and then uses general machinery to obtain the general case as a limit.
A discussion of this history of this proof can be found in Theorem 4.1 in Gardner's survey on Brunn–Minkowski.
First, we observe that there is an axis aligned hyperplane H that such that each side of H contains an entire box of A.
, so we can calculate: In this setting, both bodies can be approximated arbitrarily well by unions of disjoint axis aligned rectangles contained in their interior; this follows from general facts about the Lebesgue measure of open sets.
By using associativity and commutativity of Minkowski sum, along with the previous case, we can calculate that
The argument here is simpler than the proof via cuboids, in particular, we only need to prove the BM inequality in one dimensions.
QED The multiplicative version of Brunn–Minkowski follows from the PL inequality, by taking
We now show how the multiplicative BM-inequality implies the usual, additive version.
The Brunn–Minkowski inequality gives much insight into the geometry of high dimensional convex bodies.
For more general bodies this radius function does not appear to have a completely clear geometric interpretation beyond being the radius of the disc obtained by packing the volume of the slice as close to the origin as possible; in the case when K(x) is not a disc, the example of a hypercube shows that the average distance to the center of mass can be much larger than r(x).
Sometimes in the context of a convex geometry, the radius function has a different meaning, here we follow the terminology of this lecture.
This shows that the radius function is concave on its support, matching the intuition that a convex body does not dip into itself along any direction.
The concavity of the radius function defined in the previous section implies that that
Grunbaum's theorem can be proven using Brunn–Minkowski inequality, specifically the convexity of the Brunn–Minkowski symmetrization.
[3] Grunbaum's inequality has the following fair cake cutting interpretation.
hard,[4] limiting the usefulness of this cake cutting strategy for higher dimensional, but computationally bounded creatures.
Applications of Grunbaum's theorem also appear in convex optimization, specifically in analyzing the converge of the center of gravity method.
This agrees with the usual meaning of surface area by the Minkowski-Steiner formula.
We use this calculation to lower bound the surface area of
We apply the multiplicative form of the Brunn–Minkowski inequality to lower bound the first term by
However, the notion of surface area requires modification, see: the aforementioned notes on concentration of measure from Barvinok.
The proof of the Brunn–Minkowski theorem establishes that the function is concave in the sense that, for every pair of nonempty compact subsets A and B of Rn and every 0 ≤ t ≤ 1, For convex sets A and B of positive measure, the inequality in the theorem is strict for 0 < t < 1 unless A and B are positive homothetic, i.e. are equal up to translation and dilation by a positive factor.
, we see that we can think of the cross terms of the binomial expansion of the latter as accounting, in some fashion, for the mixed volume representation of
denotes the middle third Cantor set, then it is an exercise in analysis to show that
The Brunn–Minkowski inequality continues to be relevant to modern geometry and algebra.
For instance, there are connections to algebraic geometry,[6][7] and combinatorial versions about counting sets of points inside the integer lattice.