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We survey problems and results from combinatorial geometry in normed spaces, concentrating on problems that involve distances. These include various properties of unit-distance graphs, minimum-distance graphs, diameter graphs, as well as minimum spanning trees and Steiner minimum trees. In particular, we discuss translative kissing (or Hadwiger) nu...
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... [43] considered general two-dimensional normed spaces, and showed that the bounds 4 χ u (X 2 ) 7 hold for all X 2 . The lower bound follows since the so-called Moser spindle (Fig. 2) still occurs as a unit-distance graph for any norm, and the upper bound comes from an appropriate tiling of the plane by a hexagon of sides lengths 1/2 inscribed in the circle of radius 1/2. Chilakamarri notes that the chromatic number is exactly 4 if the unit ball is a parallelogram or a hexagon, and at most 6 if the unit ball is an ...
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Citations
... An explicit upper bound on the necessary grid size is stated in Proposition 32. While there are infinitely many 7-point sets spanning two squares that are pairwise non-similar, for all numbers n of points and m of squares there exist only finitely many equivalence classes if one uses a suitable combinatorial description, see Definition 18 and Definition 19. With this, the determination of S (n) becomes a finite computational problem. ...
... For the subsequently mentioned integer sequences see the "On-Line Encyclopedia of Integer Sequences" (OEIS) at https://oeis.org.2 This upper bound applies to all strictly convex norms, not just Euclidean distance, and can in fact be attained for certain special norms, see[20,19]. ...
How many squares are spanned by $n$ points in the plane? Here we study the corresponding maximum possible number $S_{\square}(n)$ of squares and determine the exact values for all $n\le 17$. For $18\le n\le 100$ we give lower bounds for $S_{\square}(n)$. Besides that a few preliminary structural results are obtained. For the related problem of the maximum possible number of isosceles right triangles we determine the exact values for $n\le 14$ and give lower bounds for $15\le n\le 50$.
... As far as we know, this problem has not been well studied in this space. Swanepoel [Swa18] showed that e(X ⊕ ∞ Y ) ≤ e(X)b f (Y ) for normed spaces X and Y , where b f is the finite Borsuk number. However, explicit bounds are still unknown when X and Y are Euclidean spaces and when we take an ℓ p sum instead of an ℓ ∞ sum. ...
... Note that we have the obvious lower bound e(E a ⊕ ∞ E b ) ≥ e(E a )e(E b ) = (a+1)(b+1) by taking the Cartesian product of the two equilateral sets. This lower bound actually meets the upper bound when a = 2, 3 [Swa18]. ...
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... Regarding upper bounds on the equilateral number, a classical result of Petty [Pet71] and Soltan [Sol75] shows that e(X) ≤ 2 n for any X of dimension n, with equality if and only if the unit ball of X is an affine image of the n-dimensional cube. For more background on the equilateral number see Section 3 of the survey [Swa18]. ...
For fixed $k$ we prove exponential lower bounds on the equilateral number of subspaces of $\ell_{\infty}^n$ of codimension $k$. In particular, we show that if the unit ball of a normed space of dimension $n$ is a centrally symmetric polytope with at most $\frac{4n}{3}-o(n)$ pairs of facets, then it has an equilateral set of cardinality at least $n+1$. These include subspaces of codimension $2$ of $\ell_{\infty}^{n+2}$ for $n\geq 9$ and of codimension $3$ of $\ell_{\infty}^{n+3}$ for $n\geq 15$.
... Swinnerton-Dyer [10] showed that H L (C) ≥ n 2 + n for all n-dimensional convex bodies C which is the best-known lower bound valid for all convex bodies. Zong [13] (see also [9], Sec. 2.2) conjectured that ...
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The geometry of finite-dimensional normed spaces (= Minkowski geometry) is a research topic which is related to many other fields, such as convex geometry, discrete and computational geometry, subfields of functional analysis, Finsler geometry, and optimization. For that reason, Minkowski geometry is connected with a lot of interesting and recent research directions. The main objective of this exposition is to discuss some recent progress in various ramifications of the subject area, as well as to present some open problems and research directions. We also present brief historical overviews and some basic concepts of the theory.KeywordsAngular measuresBirkhoff orthogonalityCurvatureCurvature typesDifferential geometryDistance geometry problemElementary geometryEquilateral setFermat–Torricelli problemFinsler geometryGeometric constantsHadwiger’s covering conjectureMahler’s conjectureMinkowski billiardsMinkowski geometryNormed spacesReduced bodiesViterbo’s conjectureMSC 201052A2152A1052A1552A2052A3852A4046B2046B8553A1553A3553B40
Kusner asked if n+1 points is the maximum number of points in Rn such that the ℓp distance (1<p<∞) between any two points is 1. We present an improvement to the best known upper bound when p is large in terms of n, as well as a generalization of the bound to s-distance sets. We also study equilateral sets in the ℓp sums of Euclidean spaces, deriving upper bounds on the size of an equilateral set for when p=∞, p is even, and for any 1≤p<∞.
For fixed k we prove exponential lower bounds on the equilateral number of subspaces of \(\ell _{\infty }^n\) of codimension k. In particular, we show that subspaces of codimension 2 of \(\ell _{\infty }^{n+2}\) and subspaces of codimension 3 of \(\ell _{\infty }^{n+3}\) have an equilateral set of cardinality \(n+1\) if \(n\ge 7\) and \(n\ge 12\) respectively. Moreover, the same is true for every normed space of dimension n, whose unit ball is a centrally symmetric polytope with at most \({4n}/{3}-o(n)\) pairs of facets.
