Kakutani’s conditions

Abstract

We begin with three conditions of Joichi which imply easily the Gurari-Sozonov condition (3,8):

• (12.1) Forevry 2-dimensional subspace F of E,$x \in P_F^{ - {1_0}}$and y ∈ F, x # y (i.e. every plane is centrally symmetric).

• (12.2) For every 2-dimensional subspace F of E and every$x \in P_F^{ - {1_0}}$, ${P_{{S_F}}}x = {S_F}$ . (i.e. ǁx - uǁ = ǁx - vǁ ∀u,v ∈ SF).

• (12.3) For every 2-dimensional subspace F of E,x ∈ E and 0 ≠ y ∈ PFx, sgny ∈$\in {P_{{S_F}}}x$ .