## Convex hulls of finite packs of spheres

**Goal:**
The linear pack of spheres of length n in dimension d is the convex hull of n non-overlapping balls of radius one, with centers arranged on a line segment and without gaps between the balls.
The volume of this linear pack shall be compared against the volume of the convex hull of other non-overlapping arrangements of d-dimensional balls, in the quest of finding the arrangement of minimal volume.
It is obvious that in dimension d=2, the triangular arrangement is more efficient (in terms of lower volume of its convex hull) than the linear pack.

The task of the project group will be to fix d > 2, and to find an arrangement with n as small as possible which is more efficient than the linear pack of length n.

Already in dimension d = 3, this requires some efforts, but also dimension d = 4 should be tackled. Arrangements constructed in low dimensions should be generalised for arbitrary dimension, and the volume of their convex hull computed on the machine for several values of d, in order to compare against the linear pack.

Ideally, the project group could work simultaneously on Jean-Marc Schlenker's project "Lower bounds for kissing numbers", because the expertise and software to be developed by the project group would be useful for both projects. And results on both projects could compensate for the introductory level of the tasks.

**Schedule:** to be determined.

**Persons involved:**
Advised by Alexander D. Rahm.

**Difficulty level:**
Introductory.

**Tools:**
programs in python/sage/matlab/maple/mathematica/GAP/Pari-GP or whatever the project team likes to use.

**Expected results:**
Find packs of minimal length n which are more efficient than the linear pack.