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- H. Groemer, L. J. Wallen
- 2001

We introduce for convex domains of constant width a measure of asymmetry and show that the most asymmetric domains are Reuleaux triangles. Measures of (central) symmetry, or, as we prefer, asymmetry for convex bodies have been extensively investigated, especially with respect to determining the extremal bodies. A survey of results of this kind (up to 1963)â€¦ (More)

- H. Groemer
- Discrete & Computational Geometry
- 2000

- H. Groemer
- Discrete & Computational Geometry
- 1986

This article concerns packings and coverings that are formed by the application of rigid motions to the members of a given collection K of convex bodies. There are two possibilities to construct such packings and coverings: One may permit that the convex bodies from K are used repeatedly, or one may require that these bodies should be used at most once. Inâ€¦ (More)

- H. Groemer
- J. Comb. Theory, Ser. A
- 1983

- H. Groemer
- Discrete & Computational Geometry
- 1990

It is well known that an n-dimensional convex body permits a lattice packing of density I only if it is a centrally symmetric potytope of at most 2(2" 1) facets. This article concerns itself with the associated stability problem whether a convex body that permits a packing of high density is in some sense close to such a polytope. Several inequalities thatâ€¦ (More)

- H. Groemer
- 1993

Let h be the support function of a d-dimensional convex body and a real valued function on the unit sphere. The primary objective of this note is to nd conditions on that imply that h + is again a support function. It is shown that for a large class of convex bodies such a condition can be formulated as an inequality involving and its rst and second orderâ€¦ (More)

- H. Groemer
- 2010

f(X) Ã¨ 1 is called a star body (cf. K. Mahler [l]). The function by which a star body is defined is determined uniquely. Suppose now that 5 is bounded. There exists a least number k such that for all X and Y f(X+ Y) Ã¨Hf(X)+f(Y)); this number k is called the concavity coefficient of 5. One has always k ^ 1 and furthermore k = 1 if and only if 5 is convex.â€¦ (More)

- H. Groemer
- 2004

With any given convex body we associate three numbers that exhibit, respectively, its deviation from a ball, a centrally symmetric body, and a body of constant width. Several properties of these deviation measures are studied. Then, noting that these special bodies may be defined in terms of their normals, corresponding deviation measures for normals areâ€¦ (More)

- H. Groemer
- 2010

Let [Kx, K2, â– â– â– } be a class of compact convex subsets of euclidean M-space with the property that the set of their diameters is bounded. It is shown that the sets A, can be rearranged by the application of rigid motions so as to cover the total space if and only if the sum of the volumes of all the sets A, is infinite. Also, some statements regardingâ€¦ (More)

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