The Heilbronn Problem

Place n points in a region of unit area so that the smallest triangle formed by any three of them has the largest possible area A(n).

Best known configurations for three containers, after Erich Friedman's Packing Center — with exact coordinates for download, congruence and symmetry shown on every figure, links to the proofs, and an in-browser verifier.

Squares

n points in the unit square.

Triangles

n points in a triangle of unit area.

Convex regions

n points in a convex region chosen freely, of unit area.

Best known values

Truncated to 8 decimals; ▲ marks entries proven optimal.

nsquaretriangleconvex
3 0.50000000 1.00000000 1.00000000
4 0.50000000 0.33333333 0.50000000
5 0.19245008 0.17157287 0.27639320
6 0.12500000 0.12500000 0.16666666
7 0.08385900 0.09722222 0.11111111
8 0.07237642 0.06778921 0.08000013
9 0.05487599 0.05484693 0.06406475
10 0.04653741 0.04337674 0.05199307
11 0.03703703 0.03652988 0.04255319
12 0.03259885 0.03100478 0.03921568
13 0.02701883 0.02655652 0.03093688
14 0.02430383 0.02377577 0.02783557
15 0.02110535 0.02109076 0.02456405
16 0.02052785 0.01797627 0.02227287
17 0.01648122 0.01551737 0.01873145
18 0.01443265 0.01339346 0.01823889
19 0.01267739 0.01136764 0.01452730
20 0.01085603 0.01010844 0.01326686
21 0.01081492 0.01062286 0.01195937
22 0.00945896 0.00931322 0.01066121
23 0.00881275 0.00909416 0.00954454
24 0.00797766 0.00898515 0.00884932
25 0.00724467 0.00683834 0.00794202
26 0.00652909 0.00649165 0.00840326
27 0.00590163 0.00636117 0.00780376
28 0.00563598 0.00570263 0.00638685
29 0.00561966 0.00541009 0.00607881
30 0.00544512 0.00509805 0.00549242
31 0.00510608 0.00475594 0.00513641
32 0.00469033 0.00417795 0.00488672
33 0.00414638 0.00396742 0.00461434
34 0.00400475 0.00375818 0.00441351
35 0.00370249 0.00360622 0.00405669