Glasnik Matematicki, Vol. 33, No.2 (1998), 251-265.

ON G-PSEUDO-CENTRES OF CONVEX BODIES

Maria Moszynska and Tomasz Zukowski

Abstract.   As it is well known, for every convex body A in Rⁿ there is a unique centrally symmetric kernel, that is, a centrally symmetric convex body C ⊂ A with maximal n-volume. The paper concerns G-kernels of a convex body A for any subgroup G of O(n), i.e. G-invariant convex subsets of A with maximal n-volume. We prove that only for G generated by the central symmetry σ₀ every A has a unique G-kernel. If A is strictly convex, then its G-kernel is unique for every G.

1991 Mathematics Subject Classification.   52A20, 52A38, 52A99.

Key words and phrases.   Convex body, G-invariant convex body, n-volume, G-kernel, G-pseudo-centre.