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Israel Journal of Mathematics

, Volume 159, Issue 1, pp 317–341 | Cite as

Smooth convex bodies with proportional projection functions

  • Ralph Howard
  • Daniel Hug
Article

Abstract

For a convex body K ⊂ ℝn and i ∈ {1, …, n − 1}, the function assigning to any i-dimensional subspace L of ℝn, the i-dimensional volume of the orthogonal projection of K to L, is called the i-th projection function of K. Let K, K 0 ⊂ ℝn be smooth convex bodies with boundaries of class C 2 and positive Gauss-Kronecker curvature and assume K 0 is centrally symmetric. Excluding two exceptional cases, (i, j) = (1, n − 1) and (i, j) = (n − 2, n − 1), we prove that K and K 0 are homothetic if their i-th and j-th projection functions are proportional. When K 0 is a Euclidean ball this shows that a convex body with C 2 boundary and positive Gauss-Kronecker with constant i-th and j-th projection functions is a Euclidean ball.

Keywords

Convex Body Projection Function Support Function Constant Width Euclidean Ball 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© The Hebrew University of Jerusalem 2007

Authors and Affiliations

  • Ralph Howard
    • 1
  • Daniel Hug
    • 2
  1. 1.Department of MathematicsUniversity of South CarolinaColumbiaUSA
  2. 2.Mathematisches InstitutAlbert-Ludwigs-Universität FreiburgD-79104Germany

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