Math 2.0

Convex and discrete geometry Definition

Brunn–Minkowski theory

Builds on Convex bodyMinkowski additionMixed volumeLebesgue measureBrunn–Minkowski theorem

Definition

Let n≥1n \ge 1, let 𝒦n\mathcal{K}^n be the set of convex bodies in ℝn\R^n, and let VV be the volume, that is, Lebesgue measure. Brunn–Minkowski theory is the study of the functionals on 𝒦n\mathcal{K}^n obtained from VV and Minkowski addition K+L={x+y:x∈K, y∈L}K + L = \set{x + y : x \in K, \ y \in L} together with scaling by nonnegative reals. Its basis is Minkowski's theorem: for K1,…,Km∈𝒦nK_1, \ldots, K_m \in \mathcal{K}^n and λ1,…,λm≥0\lambda_1, \ldots, \lambda_m \ge 0, V(λ1K1+⋯+λmKm)=∑i1,…,in=1mλi1⋯λinV(Ki1,…,Kin),V(\lambda_1 K_1 + \cdots + \lambda_m K_m) = \sum_{i_1, \ldots, i_n = 1}^{m} \lambda_{i_1} \cdots \lambda_{i_n} V(K_{i_1}, \ldots, K_{i_n}), where the coefficients V(Ki1,…,Kin)V(K_{i_1}, \ldots, K_{i_n}), symmetric in their arguments, are the mixed volumes. The theory studies the mixed volumes and their special cases, and inequalities between them such as the Brunn–Minkowski theorem.

Remarks

Example: take K1=KK_1 = K and K2=BK_2 = B, the unit ball. Then V(K+tB)V(K + t B) is a polynomial of degree nn in t≥0t \ge 0; this is Steiner's formula. Its coefficients are, up to binomial factors, the quermassintegrals of KK, among them the volume and a multiple of the surface area. Non-example: Lutwak's dual Brunn–Minkowski theory replaces Minkowski addition by radial addition of star bodies; it studies sections rather than projections and is a different theory. The LpL_p Brunn–Minkowski theory is an extension that replaces Minkowski addition by LpL_p addition of support functions.

Reading path

To understand this page from the ground up, read these 10 pages in order. The foundations come first.

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Borel sigma-algebra, concept, no evidence levelBorel sigma-algebraconceptCompact set, concept, no evidence levelCompact setconceptConvex body, concept, no evidence levelConvex bodyconceptEuclidean space, object, no evidence levelEuclidean spaceobjectMinkowski addition, concept, no evidence levelMinkowski additionconceptMixed volume, concept, no evidence levelMixed volumeconceptOuter measure, concept, no evidence levelOuter measureconceptLebesgue measure, object, no evidence levelLebesgue measureobjectBrunn–Minkowski theorem, statement, no evidence levelBrunn–Minkowski theor…statementBrunn–Minkowski theory, definition, no evidence levelBrunn–Minkowski theorydefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Borel sigma-algebra concept
  2. Compact set concept
  3. Convex body concept
  4. Euclidean space object
  5. Minkowski addition concept
  6. Mixed volume concept
  7. Outer measure concept
  8. Lebesgue measure object
  9. Brunn–Minkowski theorem statement
  10. Brunn–Minkowski theory definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Brunn%E2%80%93Minkowski_theorem
About this page
TagVDPTQ35JWE
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Kinddefinition
WritingW0 stub
MSC52A39, 52A40, 52A20
DefinesBrunn–Minkowski theory
UsesConvex body, Minkowski addition, Mixed volume, Lebesgue measure, Brunn–Minkowski theorem

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