Gilbert–Pollak conjecture

The conjecture that, for every finite set of points in the Euclidean plane, the minimum Steiner tree length divided by the minimum spanning tree length is at least √3/2, with equality attainable.

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Steiner tree: A shortest network connecting a given set of points, possibly using additional junctions. Its minimum length supplies the numerator in the conjectured ratio.
Edgar Gilbert: An American mathematician and information theorist who studied geometric network problems. He co-formulated the conjecture with Henry Pollak in 1968.
Equilateral triangle: A triangle with three equal sides and three equal angles. Its Steiner tree uses a central 120-degree junction and attains the ratio √3/2.
Steiner ratio in higher dimensions: The minimum Steiner-to-spanning-tree length ratio for point sets in Euclidean spaces of dimension greater than two. The planar conjecture does not settle the corresponding constants in higher dimensions.
Minimum spanning tree: A shortest cycle-free network connecting all vertices of a weighted graph. Its Euclidean length supplies the denominator against which Steiner trees are compared.
Henry Pollak: An American mathematician known for work in approximation theory and geometric optimization. He co-formulated the conjecture with Edgar Gilbert.
Steiner tree for three terminals: A shortest network connecting three points, with an added junction when all angles are below 120 degrees. The three-terminal case supplies the sharp example motivating the conjectured constant.
Euclidean minimum spanning tree: A minimum spanning tree whose edge weights are Euclidean distances between points. It is the exact baseline network used in the planar ratio.
Steiner ratio: The ratio of minimum Steiner-tree length to minimum spanning-tree length for a point set. The conjecture identifies the sharp lower bound for this quantity in the plane.
Gilbert and Pollak 1968: The paper that introduced the planar Steiner-ratio conjecture and established foundational bounds. It states the conjecture and proves the matching upper bound through a three-point example.