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Intersections and Translative Integral Formulas for Boundaries of Convex Bodies
Author(s) -
Hug Daniel,
Schätzle Reiner
Publication year - 2001
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/1522-2616(200106)226:1<99::aid-mana99>3.0.co;2-s
Subject(s) - mathematics , combinatorics , euler characteristic , homotopy , regular polygon , conjecture , convex body , euler's formula , measure (data warehouse) , pure mathematics , mathematical analysis , geometry , convex hull , database , computer science
Let K , L ⊂ ℝ n be two convex bodies with non–empty interiors and with boundaries ∂ K , ∂ L , and let χ denote the Euler characteristic as defined in singular homology theory. We prove two translative integral formulas involving boundaries of convex bodies. It is shown that the integrals of the functions t ↦ χ (∂ K ∩ (∂ L + t )) and t ↦ χ (∂ K ∩ ( L + t )), t ∈ ℝ n , with respect to an n –dimensional Haar measure of ℝ n can be expressed in terms of certain mixed volumes of K and L . In the particular case where K and L are outer parallel bodies of convex bodies at distance r > 0, the result will be deduced from a recent (local) translative integral formula for sets with positive reach. The general case follows from this and from the following (global) topological result. Let K r , L r denote the outer parallel bodies of K , L at distance r ≥ 0. Establishing a conjecture of Firey (1978), we show that the homotopy type of ∂ K r ∩ ∂ L r and ∂ K r ∩ L r , respectively, is independent of r ≥ 0 if K ° ∩ L ° ≠ ∅ and if ∂ K and ∂ L intersect almost transversally.As an immediate consequence of our translative integral formulas, we obtain a proof for two kinematic formulas which have also been conjectured by Firey .