P32 · Extremal configurations · Formed here

The smallest triangle in a quadrant

Place n points inside a quadrant (quarter-disc) of radius 1 so that the smallest triangle formed by any three of them is as large as possible.

Formal definition

ContainerThe container is a quarter-disc of radius 1: centred at the origin (0, 0), with its straight edges along the axes from 0 to 1 and the arc in the first quadrant.
SubmissionExactly n points, no three collinear
ConstraintsEvery point lies inside the container or on its boundary
ObjectiveMake the smallest triangle over all triples as large as possible; compared internally by twice the area, exactly

Getting a feel for it

Where the room for improvement is

Even spreading is not enough: no three points may come close to collinear, and near-collinearity is exactly what tidy arrangements love to do. Optima are often asymmetric and hard even to guess.

Where the frontier is

Our own variant: Heilbronn's problem in a quadrant (quarter-disc) of radius 1 was posed here, and there is no literature for it. Every n is unstudied; the standing record is all anybody knows.