Isometry groups of norms
28th September 2022, 2:30 pm – 3:30 pm
Fry Building, 2.04
There are many well-known and important examples of norms on R^n, and each has an isometry group that is a compact subgroup of GL(n,R). A classical question, going back to J Lindenstrauss and others, asks which compact linear groups can occur as the isometry group of some norm. I shall present a necessary and sufficient criterion for this which can be stated in a rather simple way in terms of the orbits of the group. One consequence is that every finite subgroup of GL(n,R) is the isometry group of a norm, an old result of Gordon and Loewy. The criterion leads to other interesting results and questions about compact Lie groups and their representations, which I shall discuss.