The Blaschke–Kakutani result characterizes inner product spaces $E$, among normed spaces of dimension at least 3, by the property that for every 2 dimensional subspace $F$ there is a norm 1 linear projection onto $F$. In this paper, we determine which closed neighborhoods $B$ of zero in a real locally convex space $E$ of dimension at least 3 have the property that for every 2 dimensional subspace $F$ there is a continuous linear projection $P$ onto $F$ with $P\left(B\right)\subseteq B$.