Geometrical Optimisation Problem Consider a sphere in a space of d dimensions. The middle point of the sphere is the origin and its radius is 1. We consider part of the space for which all co-ordinate values are larger than or equal to 0. The sphere is enclosed by a polytope. This polytope is the convex hull of n points. A second sphere encloses the polytope. This sphere has a radius d. The challenge is to optimise the point co-ordinates such that d obtains its smallest value. The table below shows the smallest d for different values of d and n. For small values of d or n the solutions have been found analytically. For other values computer programs have been used. The computation effort for these problems can be very large. Especially determining the properties of the polytope is time consuming. Moreover, the latter needs to be repeated for each improved point co-ordinate. At the moment a computer is running several big jobs. When a new solution has been found it will be added to this table.

This research is applied to optimal load combinations for analysing engineering structures.

	d=1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
n=1	1.00
2	1.00	1.41
3	1.00	1.08	1.73
4	1.00	1.04	1.41	2.00
5	1.00	1.02	1.41	1.73	2.24
6	1.00	1.01	1.22	1.53	2.00	2.45
7	1.00	1.01	1.13	1.47	1.78	2.24	2.65
8	1.00	1.01		1.41	1.78	2.00	2.45	2.83
9	1.00	1.00	1.08	1.41	1.78	1.87	2.24	2.65	3.00
10	1.00	1.00	1.08	1.41	1.63		2.08	2.45	2.83	3.16
11	1.00	1.00		1.22	1.58			2.27	2.65	3.00	3.32
12	1.00	1.00	1.06	1.22	1.58				2.45	2.83	3.16	3.46
13	1.00	1.00		1.22	1.53				2.35	2.65	3.00	3.32	3.61
14	1.00	1.00		1.18	1.53	1.73	2.24			2.52	2.83		3.46	3.74
15	1.00	1.00	1.04	1.16	1.41	1.63				2.42	2.68			3.61	3.87
16	1.00	1.00	1.04	1.16	1.41			2.45							3.74	4.00
17	1.00	1.00	1.04	1.16												3.87	4.12
18	1.00	1.00	1.04	1.16					2.65								4.00	4.24
19	1.00	1.00																4.12	4.36
20	1.00	1.00								2.83									4.24	4.47

	d=2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30
n=21	1.00	1.03																	4.36	4.58
22	1.00	1.03								*.**										4.47	4.69
23	1.00	1.03																			4.58	4.80
24	1.00										*.**											4.69	4.90
25	1.00																						4.80	5.00
26	1.00				1.41							3.32												4.90	5.10
27	1.00																								5.00	5.20
28	1.00	1.02											*.**													5.10	5.29
29	1.00	1.02																									5.20	5.39
30	1.00	1.02		1.20										*.**														5.29	5.48
31	1.00	1.02		1.18																									5.39
32	1.00			1.18											*.**
33	1.00			1.18
34	1.00			1.18												*.**
35	1.00			1.18
36	1.00			1.18			2.00										*.**
37	1.00
38	1.00	1.01																*.**
39	1.00	1.01
40	1.00	1.01	1.02	1.02	1.02	1.02	1.02	1.02	1.02	1.02	1.02	1.02	1.02	1.02					*.**

	d=2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30
n=41	1.00	1.01
42	1.00	1.01				1.41														*.**
43	1.00	1.01
44	1.00	1.01																			*.**
45	1.00	1.01
46	1.00	1.01																				*.**
47	1.00	1.01
48	1.00	1.01																					*.**
49	1.00	1.01
50	1.00	1.01																						*.**
51	1.00	1.01
52	1.00	1.01																							*.**
53	1.00	1.01
54	1.00	1.01																								*.**
55	1.00	1.01
56	1.00	1.01																									*.**
57	1.00	1.01
58	1.00	1.01																										*.**
59	1.00	1.01
60	1.00	1.01	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00										*.**