Discrete Geometry

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6 domain-specific abstractions whose origin domain is Discrete Geometry.

• Danzer set — A set of points in Euclidean space that intersects every convex body of unit volume, studied through the unresolved question of whether bounded-density examples exist.
• Gilbert–Pollak conjecture — The assertion, now a theorem under accepted proof, that a Euclidean minimum spanning tree is at most 2/√3 times as long as a Steiner minimum tree on the same planar terminals.
• Kissing number — The maximum number of nonoverlapping congruent spheres that can simultaneously touch one congruent central sphere in a specified space or dimension.
• Opaque set — A set of planar curves or segments intersecting every line that crosses a specified convex body.
• Perfect rectangle — A rectangle tiled exactly by finitely many squares whose side lengths are all distinct.
• Tetrahedron packing — The geometric optimization problem of arranging congruent regular tetrahedra without overlapping so as to maximize the fraction of three-dimensional space they occupy.