An upper bound on the volume of the symmetric difference of a body and a congruent copy
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D. Schymura
Abstract
Let A be a bounded subset of ℝd for some d ≥ 2. We give an upper bound on the volume of the symmetric difference of A and ƒ(A) where f is a translation, a rotation, or the composition of both, a rigid motion.
We bound the volume of the symmetric difference of A and f(A) in terms of the (d - 1)- dimensional volume of the boundary of A and the maximal distance of a boundary point to its image under ƒ. The boundary is measured by the (d - 1)-dimensional Hausdorff measure, which matches the surface area for sufficiently nice sets. In the case of translations, our bound is sharp. In the case of rotations, we get a sharp bound under the assumption that the boundary is sufficiently nice.
The motivation to study these bounds comes from shape matching.
©2014 by Walter de Gruyter Berlin/Boston
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- On a reverse Petty projection inequality for projections of convex bodies
- Generic properties of homogeneous Ricci solitons
- On the classification of convex lattice polytopes (II)
- Idempotent tropical matrices and finite metric spaces
- Legendre duality on hypersurfaces in Kähler manifolds
- An upper bound on the volume of the symmetric difference of a body and a congruent copy
- Cubic tessellations of the didicosm
- Isometries of complemented sub-Riemannian manifolds
- The isomorphism problem for linear representations and their graphs The isomorphism problem for linear representations and their graphs
- Translation ovoids of unitary polar spaces
Artikel in diesem Heft
- Masthead
- Geometric structures over non-reductive homogeneous 4-spaces
- On a reverse Petty projection inequality for projections of convex bodies
- Generic properties of homogeneous Ricci solitons
- On the classification of convex lattice polytopes (II)
- Idempotent tropical matrices and finite metric spaces
- Legendre duality on hypersurfaces in Kähler manifolds
- An upper bound on the volume of the symmetric difference of a body and a congruent copy
- Cubic tessellations of the didicosm
- Isometries of complemented sub-Riemannian manifolds
- The isomorphism problem for linear representations and their graphs The isomorphism problem for linear representations and their graphs
- Translation ovoids of unitary polar spaces
