Brunn–Minkowski theorem

Brunn–Minkowski theorem In mathematics, the Brunn–Minkowski theorem (or Brunn–Minkowski inequality) is an inequality relating the volumes (or more generally Lebesgue measures) of compact subsets of Euclidean space. 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).

Contents 1 Statement 1.1 Multiplicative version 2 On the hypothesis 2.1 Measurability 2.2 Non-emptiness 3 Proofs 4 Important corollaries 4.1 Concavity of the radius function (Brunn's theorem) 4.2 Brunn–Minkowski symmetrization of a convex body 4.3 Grunbaum's theorem 4.4 Isoperimetric inequality 4.5 Applications to inequalities between mixed volumes 4.6 Concentration of measure on the sphere and other strictly convex surfaces 5 Remarks 6 Examples 6.1 Rounded cubes 6.2 Examples where the lower bound is loose 7 Connections to other parts of mathematics 8 See also 9 References 10 References Statement Let n ≥ 1 and let μ denote the Lebesgue measure on Rn. Let A and B be two nonempty compact subsets of Rn. Then the following inequality holds: {displaystyle [mu (A+B)]^{1/n}geq [mu (A)]^{1/n}+[mu (B)]^{1/n},} where A + B denotes the Minkowski sum: {displaystyle A+B:={,a+bin mathbb {R} ^{n}mid ain A, bin B,}.} The theorem is also true in the setting where {textstyle A,B,A+B} are only assumed to be measurable and non-empty.[1] Multiplicative version The multiplicative form of Brunn–Minkowski inequality states that {textstyle mu (lambda A+(1-lambda )B)geq mu (A)^{lambda }mu (B)^{1-lambda }} for all {textstyle lambda in [0,1]} .

The Brunn–Minkowski inequality is equivalent to the multiplicative version.

In one direction, use the inequality {textstyle lambda x+(1-lambda )ygeq x^{lambda }y^{1-lambda }} (Young's inequality for products), which holds for {textstyle x,ygeq 0,lambda in [0,1]} . In particular, {textstyle mu (lambda A+(1-lambda )B)geq (lambda mu (A)^{1/n}+(1-lambda )mu (B)^{1/n})^{n}geq mu (A)^{lambda }mu (B)^{1-lambda }} .

Conversely, using the multiplicative form, we find {textstyle mu (A+B)=mu (lambda {frac {A}{lambda }}+(1-lambda ){frac {B}{1-lambda }})geq {frac {mu (A)^{lambda }mu (B)^{1-lambda }}{lambda ^{nlambda }(1-lambda )^{n(1-lambda )}}}} The right side is maximized at {displaystyle lambda =1-{frac {1}{1+e^{C}}},C={frac {1}{n}}ln {frac {mu (B)}{mu (A)}}} , which gives {textstyle mu (A+B)geq (mu (A)^{1/n}+mu (B)^{1/n})^{n}} .

The Prékopa–Leindler inequality is a functional generalization of this version of Brunn–Minkowski.

On the hypothesis Measurability It is possible for {textstyle A,B} to be Lebesgue measurable and {textstyle A+B} to not be; a counter example can be found in "Measure zero sets with non-measurable sum." On the other hand, if {textstyle A,B} are Borel measurable, then {textstyle A+B} is the continuous image of the Borel set {textstyle Atimes B} , so analytic and thus measurable. See the discussion in Gardner's survey for more on this, as well as ways to avoid measurability hypothesis.

We note that in the case that A and B are compact, so is A + B, being the image of the compact set {textstyle Atimes B} under the continuous addition map : {textstyle +:mathbb {R} ^{n}times mathbb {R} ^{n}to mathbb {R} ^{n}} , so the measurability conditions are easy to verify.

Non-emptiness The condition that {textstyle A,B} are both non-empty is clearly necessary. This condition is not part of the multiplicative versions of BM stated below.

Proofs We give two well known proofs of Brunn–Minkowski.

show Geometric proof via cuboids and measure theory show Proof as a corollary of the Prékopa–Leindler inequality Important corollaries The Brunn–Minkowski inequality gives much insight into the geometry of high dimensional convex bodies. In this section we sketch a few of those insights.

Concavity of the radius function (Brunn's theorem) Consider a convex body {textstyle Ksubseteq mathbb {R} ^{n}} . Let {textstyle K(x)=Kcap {x_{1}=x}} be vertical slices of K. Define {textstyle r(x)=mu (K(x))^{frac {1}{n-1}}} to be the radius function; if the slices of K are discs, then r(x) gives the radius of the disc K(x), up to a constant. 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). We note that sometimes in the context of a convex geometry, the radius function has a different meaning, here we follow the terminology of this lecture.

By convexity of K, we have that {textstyle K(lambda x+(1-lambda )y)supseteq lambda K(x)+(1-lambda )K(y)} . Applying the Brunn–Minkowski inequality gives {textstyle r(K(lambda x+(1-lambda )y))geq lambda r(K(x))+(1-lambda )r(K(y))} , provided {textstyle K(x)not =emptyset ,K(y)not =emptyset } . 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. This result is sometimes known as Brunn's theorem.

Brunn–Minkowski symmetrization of a convex body Again consider a convex body {textstyle K} . Fix some line {textstyle l} and for each {textstyle tin l} let {textstyle H_{t}} denote the affine hyperplane orthogonal to {textstyle l} that passes through {textstyle t} . Define, {textstyle r(t)=Vol(Kcap H_{t})} ; as discussed in the previous section, this function is concave. Now, let {textstyle K'=bigcup _{tin l,Kcap H_{t}not =emptyset }B(t,r(t))cap H_{t}} . That is, {textstyle K'} is obtained from {textstyle K} by replacing each slice {textstyle H_{t}cap K} with a disc of the same {textstyle (n-1)} -dimensional volume centered {textstyle l} inside of {textstyle H_{t}} . The concavity of the radius function defined in the previous section implies that that {textstyle K'} is convex. This construction is called the Brunn–Minkowski symmetrization.

Grunbaum's theorem Theorem (Grunbaum's theorem[citation needed]): Consider a convex body {textstyle Ksubseteq mathbb {R} ^{n}} . Let {textstyle H} be any half-space containing the center of mass of {textstyle K} ; that is, the expected location of a uniform point sampled from {textstyle K.} Then {textstyle mu (Hcap K)geq ({frac {n}{n+1}})^{n}mu (K)geq {frac {1}{e}}mu (K)} .

Grunbaum's theorem can be proven using Brunn–Minkowski inequality, specifically the convexity of the Brunn–Minkowski symmetrization[citation needed]. See these lecture notes for a proof sketch.

Grunbaum's inequality has the following fair cake cutting interpretation. Suppose two players are playing a game of cutting up an {textstyle n} dimensional, convex cake. Player 1 chooses a point in the cake, and player two chooses a hyperplane to cut the cake along. Player 1 then receives the cut of the cake containing his point. Grunbaum's theorem implies that if player 1 chooses the center of mass, then the worst that an adversarial player 2 can do is give him a piece of cake with volume at least a {textstyle 1/e} fraction of the total. In dimensions 2 and 3, the most common dimensions for cakes, the bounds given by the theorem are approximately {textstyle .444,.42} respectively. Note, however, that in {textstyle n} dimensions, calculating the centroid is {textstyle #P} hard[citation needed], 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. See theorem 2.1 in these notes.

Isoperimetric inequality Let {textstyle B=B(0,1)={xin mathbb {R} ^{n}:||x||_{2}leq 1}} denote the unit ball. For a convex body, K, let {textstyle S(K)=lim _{epsilon to 0}{frac {mu (K+epsilon B)-mu (K)}{epsilon }}} define its surface area. This agrees with the usual meaning of surface area by the Minkowski-Steiner formula. Consider the function {textstyle c(X)={frac {mu (K)^{1/n}}{S(K)^{1/(n-1)}}}} . The isoperimetric inequality states that this is maximized on Euclidean balls.

show Proof of isoperimetric inequality via Brunn–Minkowski Applications to inequalities between mixed volumes The Brunn–Minkowski inequality can be used to deduce the following inequality {textstyle V(K,ldots ,K,L)^{n}geq V(K)^{n-1}V(L)} , where the {textstyle V(K,ldots ,K,L)} term is a mixed-volume. Equality holds iff K,L are homothetic. (See theorem 3.4.3 in Hug and Weil's course on convex geometry.) show Proof Concentration of measure on the sphere and other strictly convex surfaces We prove the following theorem on concentration of measure, following notes by Barvinok and notes by Lap Chi Lau. See also Concentration of measure#Concentration on the sphere.

Theorem: Let {textstyle S} be the unit sphere in {textstyle mathbb {R} ^{n}} . Let {textstyle Xsubseteq S} . Define {textstyle X_{epsilon }={zin S:d(z,X)leq epsilon }} , where d refers to the Euclidean distance in {textstyle mathbb {R} ^{n}} . Let {textstyle nu } denote the surface area on the sphere. Then, for any {textstyle epsilon in (0,1]} we have that {textstyle {frac {nu (X_{epsilon })}{nu (S)}}geq 1-{frac {nu (S)}{nu (X)}}e^{-{frac {nepsilon ^{2}}{4}}}} .

show Proof Version of this result hold also for so-called strictly convex surfaces, where the result depends on the modulus of convexity. However, the notion of surface area requires modification, see: the aforementioned notes on concentration of measure from Barvinok.

Remarks The proof of the Brunn–Minkowski theorem establishes that the function {displaystyle Amapsto [mu (A)]^{1/n}} is concave in the sense that, for every pair of nonempty compact subsets A and B of Rn and every 0 ≤ t ≤ 1, {displaystyle left[mu (tA+(1-t)B)right]^{1/n}geq t[mu (A)]^{1/n}+(1-t)[mu (B)]^{1/n}.} 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. Examples Rounded cubes It is instructive to consider the case where {textstyle A} an {textstyle ltimes l} square in the plane, and {textstyle B} a ball of radius {textstyle epsilon } . In this case, {textstyle A+B} is a rounded square, and its volume can be accounted for as the four rounded quarter circles of radius {textstyle epsilon } , the four rectangles of dimensions {textstyle ltimes epsilon } along the sides, and the original square. Thus, {textstyle mu (A+B)=l^{2}+4epsilon l+{frac {4}{4}}pi epsilon ^{2}=mu (A)+4epsilon l+mu (B)geq mu (A)+2{sqrt {pi }}epsilon l+mu (B)=mu (A)+2{sqrt {mu (A)mu (B)}}+mu (B)=(mu (A)^{1/2}+mu (B)^{1/2})^{2}} . This example also hints at the theory of mixed-volumes, since the terms that appear in the expansion of the volume of {textstyle A+B} correspond to the differently dimensional pieces of A. In particular, if we rewrite Brunn–Minkowski as {textstyle mu (A+B)geq (mu (A)^{1/n}+mu (B)^{1/n})^{n}} , 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 {textstyle mu (A+B)=V(A,ldots ,A)+nV(B,A,ldots ,A)+ldots +{n choose j}V(B,ldots ,B,A,ldots ,A)+ldots nV(B,ldots ,B,A)+mu (B)} . This same phenomenon can also be seen for the sum of an n-dimensional {textstyle ltimes l} box and a ball of radius {textstyle epsilon } , where the cross terms in {textstyle (mu (A)^{1/n}+mu (B)^{1/n})^{n}} , up to constants, account for the mixed volumes. This is made precise for the first mixed volume in the section above on the applications to mixed volumes. Examples where the lower bound is loose The left-hand side of the BM inequality can in general be much larger than the right side. For instance, we can take X to be the x-axis, and Y the y-axis inside the plane; then each has measure zero but the sum has infinite measure. Another example is given by the Cantor set. If {textstyle C} denotes the middle third Cantor set, then it is an exercise in analysis to show that {textstyle C+C=[0,2]} . Connections to other parts of mathematics The Brunn–Minkowski inequality continues to be relevant to modern geometry and algebra. For instance, there are connections to algebraic geometry,[2][3] and combinatorial versions about counting sets of points inside the integer lattice.[4] See also Isoperimetric inequality Milman's reverse Brunn–Minkowski inequality Minkowski–Steiner formula Prékopa–Leindler inequality Vitale's random Brunn–Minkowski inequality Mixed volume References Brunn, H. (1887). "Über Ovale und Eiflächen". Inaugural Dissertation, München. Fenchel, Werner; Bonnesen, Tommy (1934). Theorie der konvexen Körper. Ergebnisse der Mathematik und ihrer Grenzgebiete. Vol. 3. Berlin: 1. Verlag von Julius Springer. Fenchel, Werner; Bonnesen, Tommy (1987). Theory of convex bodies. Moscow, Idaho: L. Boron, C. Christenson and B. Smith. BCS Associates. ISBN 9780914351023. Dacorogna, Bernard (2004). Introduction to the Calculus of Variations. London: Imperial College Press. ISBN 1-86094-508-2. Heinrich Guggenheimer (1977) Applicable Geometry, page 146, Krieger, Huntington ISBN 0-88275-368-1 . Lyusternik, Lazar A. (1935). "Die Brunn–Minkowskische Ungleichnung für beliebige messbare Mengen". Comptes Rendus de l'Académie des Sciences de l'URSS. Nouvelle Série. III: 55–58. Minkowski, Hermann (1896). Geometrie der Zahlen. Leipzig: Teubner. Ruzsa, Imre Z. (1997). "The Brunn–Minkowski inequality and nonconvex sets". Geometriae Dedicata. Vol. 67, no. 3. pp. 337–348. doi:10.1023/A:1004958110076. MR 1475877. Rolf Schneider, Convex bodies: the Brunn–Minkowski theory, Cambridge University Press, Cambridge, 1993. References ^ Gardner, Richard J. (2002). "The Brunn–Minkowski inequality". Bull. Amer. Math. Soc. (N.S.) 39 (3): pp. 355–405 (electronic). doi:10.1090/S0273-0979-02-00941-2. ISSN 0273-0979. ^ GROMOV, M. (1990). "CONVEX SETS AND KÄHLER MANIFOLDS". Advances in Differential Geometry and Topology. WORLD SCIENTIFIC. pp. 1–38. doi:10.1142/9789814439381_0001. ISBN 978-981-02-0494-5. ^ Neeb, Karl-Hermann (2015-10-12). "Kaehler Geometry, Momentum Maps and Convex Sets". arXiv:1510.03289v1 [math.SG]. ^ Hernández Cifre, María A.; Iglesias, David; Nicolás, Jesús Yepes (2018). "On a Discrete Brunn--Minkowski Type Inequality". SIAM Journal on Discrete Mathematics. Society for Industrial & Applied Mathematics (SIAM). 32 (3): 1840–1856. doi:10.1137/18m1166067. ISSN 0895-4801. Categories: Theorems in measure theoryTheorems in convex geometryCalculus of variationsGeometric inequalitiesSumsetsHermann Minkowski

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