Jump to Survey Papers and Exposition below.

Research Papers

Favourite distances in 3-space
Submitted.
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Ordinary hyperspheres and spherical curves
With Aaron Lin. Submitted.
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Shortest directed networks in the plane
With Alastair Maxwell. Submitted.
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On sets defining few ordinary hyperplanes
With Aaron Lin. Submitted.
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Ordinary planes, coplanar quadruples, and space quartics
With Aaron Lin. Journal of the LMS, to appear.
abstract   pdf   doi

Embedding graphs in Euclidean space
With Nóra Frankl and Andrey Kupavskii. Submitted.
abstract   pdf

Bounding the size of an almost-equidistant set in Euclidean space
With Andrey Kupavskii and Nabil H. Mustafa. Combinatorics, Probability and Computing 28 (2019), 280–286.
abstract   pdf   doi

Almost-equidistant sets
With Martin Balko, Attila Pór, Manfred Scheucher and Pavel Valtr. Submitted.
abstract   pdf

Arrangements of homothets of a convex body II
With Márton Naszódi. Contributions to Discrete Mathematics 13 (2018), 116–123.
abstract   pdf   doi (open access)

On sets defining few ordinary circles
With Aaron Lin, Mehdi Makhul, Hossein Nassajian Mojarrad, Josef Schicho and Frank de Zeeuw. Discrete & Computational Geometry 59 (2018), 59–87.
abstract   pdf   doi (open access)

Arrangements of homothets of a convex body
With Márton Naszódi and János Pach. Mathematika 63 (2017), 696–710.
abstract   pdf   doi

Approximate Euclidean Steiner trees
With Charl Ras and Doreen Thomas. Journal of Optimization Theory and Applications 172 (2017), 845–873.
abstract   pdf   doi (open access)

Sphere-of-influence graphs in normed spaces
With Márton Naszódi and János Pach. In: Discrete Geometry and Symmetry, ed. Marston D. E. Conder and Asia Ivić Weiss, Springer Proceedings in Mathematics and Statistics 234, Springer 2018, pp. 293–296
abstract   pdf   doi

Sets of unit vectors with small subset sums
Transactions of the American Mathematical Society 368 (2016), 7153–7188.
abstract   pdf   doi

Double-normal pairs in the plane and on the sphere
With János Pach. Beiträge zur Algebra und Geometrie 56 (2015), 423–438.
abstract   pdf   doi

Double-normal pairs in space
With János Pach. Mathematika 61 (2015), 259–272.
abstract   pdf   doi

Generalised k–Steiner Tree Problems in Normed Planes
With M. Brazil, C. Ras and D. A. Thomas. Algorithmica 71 (2015), 66–86.
abstract   pdf   ps   doi

Equilateral sets and a Schütte Theorem for the 4-norm
Canadian Mathematical Bulletin 57 (2014), 640–647.
abstract   pdf   doi

The centroid as an estimate for the quadratic min power centre
With M. Brazil, C. J. Ras and D. A. Thomas. 21st International Symposium on Mathematical Theory of Networks and Systems, 7th–11th July 2014, Groningen. pp. 800–803.
fulltext pdf

Maximal equilateral sets
With Rafael Villa. Discrete & Computational Geometry 50 (2013), 354–373.
abstract   pdf   ps   doi

Favorite distances in high dimensions
In: Thirty Essays in Geometric Graph Theory, ed. J. Pach, Algorithms and Combinatorics 29, Springer 2013. pp. 499–520.
abstract   pdf   ps   doi

The Gilbert arborescence problem
With M. G. Volz, M. Brazil, C. Ras and D. A. Thomas. Networks 61 (2013), 238–247.
abstract   pdf   ps   doi

Midpoint sets contained in the unit sphere of a normed space
Studia Scientiarum Mathematicarum Hungarica 48 (2011), 180–192.
abstract   pdf   ps   doi

Embedding a Latin square with transversal into a projective space
With L. M. Pretorius. Journal of Combinatorial Theory, Ser. A 118 (2011), 1674–1683.
abstract   pdf   ps   doi

Large convexly independent subsets of Minkowski sums
With P. Valtr. Electronic Journal of Combinatorics 17 (2010), #R146.
abstract   pdf   ps   fulltext pdf

Designing optimal flow networks
With M. G. Volz, M. Brazil, and D. A. Thomas. Proceedings of the World Congress on Engineering 2009 Vol II, WCE 2009, 1st–3rd July 2009, London, U.K. pp. 1235–1240.
abstract   pdf   ps   fulltext pdf

Absorbing angles, Steiner minimal trees and antipodality
With H. Martini and P. O. de Wet. Journal of Optimization Theory and Applications, 143 (2009), 149–157.
abstract   pdf   ps   doi

Simultaneous packing and covering in sequence spaces
Discrete & Computational Geometry 42 (2009), 335–340.
abstract   pdf   ps   doi

Large antipodal families
With B. Csikós, G. Kiss, and P. O. de Wet. Periodica Mathematica Hungarica 58 (2009), 129–138.
doi

Unit distances and diameters in Euclidean spaces
Discrete & Computational Geometry 41 (2009), 1–27.
abstract   pdf   ps   doi

The Sylvester–Gallai theorem, colourings and algebra
With L. M. Pretorius. Discrete Mathematics 309 (2009), 385–399.
abstract   pdf   ps   doi

A new proof of Vázsonyi's conjecture
Journal of Combinatorial Theory, Ser. A 115 (2008), 888–892.
abstract   pdf   ps   doi

Elementary incidence theorems for complex numbers and quaternions
With József Solymosi. SIAM Journal on Discrete Mathematics 22 (2008), 1145–1148.
abstract   pdf   ps   doi

A lower bound for the equilateral number of normed spaces
With Rafael Villa. Proceedings of the American Mathematical Society 136 (2008), 127–131.
abstract   pdf   ps   doi

The local Steiner problem in finite–dimensional normed spaces
Discrete & Computational Geometry 37 (2007), 419–442.
abstract   pdf   ps   doi

A generalised Sylvester–Gallai theorem
With Lou M. Pretorius. Die Suid–Afrikaanse Tydskrif vir Wetenskap en Tegnologie 26 (2007), 8–13. (Afrikaans. English Summary).
pdf

Upper bounds for edge–antipodal and subequilateral polytopes
Periodica Mathematica Hungarica 54 (2007), 99–106.
abstract   pdf   ps   doi

Three–dimensional antipodal and norm–equilateral sets
With Achill Schürmann. Pacific Journal of Mathematics 228 (2006), 349–370.
abstract   pdf   ps   doi

Blocking sets in small finite linear spaces
With L.M. Pretorius. Ars Combinatoria 80 (2006) 275–315.
abstract   pdf   ps

Sylvester–Gallai Theorems for Complex Numbers and Quaternions
With Noam Elkies and Lou M. Pretorius. Discrete & Computational Geometry 35 (2006), 361–373.
abstract   pdf   ps   doi

Low–degree minimal spanning trees in normed spaces
With H. Martini. Applied Mathematics Letters 19 (2006), 122–125.
doi

Quantitative illumination of convex bodies and vertex degrees of geometric Steiner minimal trees
Mathematika 52 (2005), 47–52.
abstract   pdf   ps   doi

Largest family without ABCD
With A. De Bonis and G.O.H. Katona. Journal of Combinatorial Theory, Ser. A. 111 (2005), 331–336.
abstract   pdf   ps   doi

A problem of Kusner on equilateral sets
Archiv der Mathematik (Basel) 83 (2004), 164–170.
abstract   pdf   ps   doi

Equiframed curves — a generalization of Radon curves
With H. Martini. Monatshefte für Mathematik 141 (2004), 301–314.
doi

Non–planar simplices are not reduced
With H. Martini. Publicationes Mathematicae Debrecen 64 (2004), 101–106.
abstract   pdf

The unit distance problem on spheres
With P. Valtr. In: Towards a Theory of Geometric Graphs, Contemporary Mathematics 342, ed. János Pach, American Mathematical Society, 2004.
pdf

An algorithmic proof of the Motzkin–Rabin theorem
With L.M. Pretorius. American Mathematical Monthly 111 no. 3 (2004), 245–251.
doi

Helly–type theorems for homothets of planar convex curves
Proceedings of the American Mathematical Society 131 (2003), 921–932.
doi

Independence numbers of planar contact graphs
Discrete & Computational Geometry 28 (2002), 649–670.
doi

The Fermat–Torricelli problem in normed planes and spaces
With H. Martini and G. Weiß. Journal of Optimization Theory and Applications 115 (2002), 283–314.
abstract   pdf   ps   doi

Helly–type theorems for polygonal curves
Discrete Mathematics 254 (2002), 527–537.
doi

Triangles of extremal area or perimeter in a finite planar point set
With Peter Braß and Günter Rote. Discrete & Computational Geometry 26 (2001), 51–58.
doi   journal

Sets of unit vectors with small pairwise sums
Quaestiones Mathematicae 23 (2000), 383–388.
abstract   pdf   ps   doi   journal

The local Steiner problem in normed planes
Networks 36 (2000), 104–113.
doi

On the existence of shortest directed networks
Journal of Combinatorial Mathematics and Combinatorial Computing 33 (2000), 97–102.
abstract   pdf   ps

Gaps in Convex Disc Packings with an Application to 1–Steiner Minimum Trees
Monatshefte für Mathematik 129 (2000), 217–226.
doi

Balancing unit vectors
Journal of Combinatorial Theory, Series A 89 (2000), 105–112.
abstract   pdf   ps   doi

Partitions of sets in metric and normed spaces induced by concentric spheres and parallel hyperplanes
Suid–Afrikaanse Tydskrif vir Wetenskap en Tegnologie 18 (1999), 116–119 (Afrikaans. English Summary).

Helly–type Theorems for Hollow Axis–aligned Boxes
Proceedings of the American Mathematical Society 127 (1999), 2155–2162.
abstract   pdf   ps   doi

New lower bounds for the Hadwiger numbers of p balls for p < 2
Applied Mathematics Letters 12 (1999), 57–60.
doi

Cardinalities of k–distance sets in Minkowski spaces
Discrete Mathematics 197/198 (1999), 759–767.
abstract   pdf   ps   doi

Vertex degrees of Steiner Minimal Trees in pd and other smooth Minkowski spaces
Discrete & Computational Geometry 21 (1999), 437–447.
abstract   pdf   ps   doi

Extremal Problems in Minkowski Space related to Minimal Networks
Proceedings of the American Mathematical Society 124 (1996), 2513–2518.
abstract   pdf   ps   doi

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Survey Papers and Exposition

Outer linear measure of connected sets via Steiner trees
Submitted.
abstract   pdf

Combinatorial distance geometry in normed spaces
New Trends in Intuitive Geometry, Bolyai Soc. Math. Studies 27, Springer, 2018, pp. 407–458.
abstract   pdf   doi

Geometry where direction matters---or does it?
With Horst Martini and Margarita Spirova.
Mathematical Intelligencer 33 (2011), 115–125.
doi

Triangle–free minimum distance graphs in the plane
Geombinatorics 19 (2009), 28–30.
pdf

Antinorms and Radon curves
With H. Martini. Aequationes Mathematicae 72 (2006), 110–138.
abstract   pdf   ps   doi

Equilateral sets in finite–dimensional normed spaces
In: Seminar of Mathematical Analysis, eds. Daniel Girela Álvarez, Genaro López Acedo, Rafael Villa Caros. Secretariado de Publicationes, Universidad de Sevilla, Seville, 2004, pp. 195–237.
abstract   pdf   ps

The Geometry of Minkowski Spaces — a survey. Part II
With H. Martini. Expositiones Mathematicae 22 (2004), 93–144.
doi

Some location problems in normed linear spaces
With H. Martini and G. Weiß. Dresden Symposium Geometrie: konstruktiv & kinematisch, 27th February–1st March 2003, Dresden, Germany.

Generalized Convexity Notions and Combinatorial Geometry
With H. Martini. Congressus Numerantium 164 (2003), 65–93.

The geometry of Minkowski spaces — a survey. Part I
With H. Martini and G. Weiß. Expositiones Mathematicae 19 (2001), 97–142.
abstract   pdf   ps   doi

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