64 research outputs found
Extremal Problems in Minkowski Space related to Minimal Networks
We solve the following problem of Z. F\"uredi, J. C. Lagarias and F. Morgan
[FLM]: Is there an upper bound polynomial in for the largest cardinality of
a set S of unit vectors in an n-dimensional Minkowski space (or Banach space)
such that the sum of any subset has norm less than 1? We prove that |S|\leq 2n
and that equality holds iff the space is linearly isometric to \ell^n_\infty,
the space with an n-cube as unit ball. We also remark on similar questions
raised in [FLM] that arose out of the study of singularities in
length-minimizing networks in Minkowski spaces.Comment: 6 pages. 11-year old paper. Implicit question in the last sentence
has been answered in Discrete & Computational Geometry 21 (1999) 437-44
Helly-type Theorems for Hollow Axis-aligned Boxes
A hollow axis-aligned box is the boundary of the cartesian product of
compact intervals in R^d. We show that for d\geq 3, if any 2^d of a collection
of hollow axis-aligned boxes have non-empty intersection, then the whole
collection has non-empty intersection; and if any 5 of a collection of hollow
axis-aligned rectangles in R^2 have non-empty intersection, then the whole
collection has non-empty intersection. The values 2^d for d\geq 3 and 5 for d=2
are the best possible in general. We also characterize the collections of
hollow boxes which would be counterexamples if 2^d were lowered to 2^d-1, and 5
to 4, respectively.Comment: 7 pages. Old paper from 199
Equilateral sets and a Sch\"utte Theorem for the 4-norm
A well-known theorem of Sch\"utte (1963) gives a sharp lower bound for the
ratio between the maximum distance and minimum distance between n+2 points in
n-dimensional Euclidean space. In this brief note we adapt B\'ar\'any's elegant
proof of this theorem to the space . This gives a new proof that the
largest cardinality of an equilateral set in is n+1, and gives a
constructive bound for an interval of values of p
close to 4 for which it is guaranteed that the largest cardinality of an
equilateral set in is n+1.Comment: 7 pages. Some technical details adde
The number of small-degree vertices in matchstick graphs
A matchstick graph is a crossing-free unit-distance graph in the plane.
Harborth (1981) proposed the problem of determining whether there exists a
matchstick graph in which every vertex has degree exactly . In 1982,
Blokhuis gave a proof of non-existence. A shorter proof was found by Kurz and
Pinchasi (2011) using a charging method. We combine their method with the
isoperimetric inequality to show that there are vertices in
a matchstick graph on vertices that are of degree at most , which is
asymptotically tight
ARRANGEMENTS OF HOMOTHETS OF A CONVEX BODY II
A family of homothets of an o-symmetric convex body K in d-dimensional Euclidean space is called a Minkowski arrangement if no homothet contains the center of any other homothet in its interior. We show that any pairwise intersecting Minkowski arrangement of a d-dimensional convex body has at most 2 . 3(d) members. This improves a result of Polyanskii (Discrete Mathematics 340 (2017), 1950-1956). Using similar ideas, we also give a proof the following result of Polyan- skii: Let , K-1, ... ,K-n be a sequence of homothets of the o-symmetric convex body K, such that for any i < j, the center of K-j lies on the boundary of K-i. Then n = O(3(d)d)
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