A codimension-one subspace can admit many norm-preserving Hahn-Banach extensions

For $(t,t)\in M$ one has $\mathrm{\mid }{f}_0(t,t)\mathrm{\mid }=\mathrm{\mid }t\mathrm{\mid }=\mathrm{\parallel }(t,t){\mathrm{\parallel }}_{\mathrm{\infty }},$ so $\mathrm{\parallel }{f}_0\mathrm{\parallel }=1$. Also every $(a,b)\in {\mathbb{R}}^2$ decomposes as $(a,b)=\frac{a+b}{2}(1,1)+\frac{a-b}{2}(1,-1),$ so $X=M\oplus \mathbb{R}(1,-1)$.

If $F$ is a linear extension of $f_0$ and $c:=F(1,-1)$, then step 1.1 forces $F(a,b)=\frac{a+b}{2}F(1,1)+\frac{a-b}{2}F(1,-1)=\frac{1+c}{2}a+\frac{1-c}{2}b\mathrm{.}$ Conversely, the displayed formula defines a linear functional extending $f_0$.

Let $\phi (a,b):=\alpha a+\beta b$ on $({\mathbb{R}}^2,\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{\mathrm{\infty }})$. If $\mathrm{\parallel }(a,b){\mathrm{\parallel }}_{\mathrm{\infty }}\le 1$, then $\mathrm{\mid }\phi (a,b)\mathrm{\mid }\le \mathrm{\mid }\alpha \mathrm{\mid }\mathrm{\mid }a\mathrm{\mid }+\mathrm{\mid }\beta \mathrm{\mid }\mathrm{\mid }b\mathrm{\mid }\le \mathrm{\mid }\alpha \mathrm{\mid }+\mathrm{\mid }\beta \mathrm{\mid }\mathrm{.}$ Choosing $a=\mathrm{sgn}\alpha$ and $b=\mathrm{sgn}\beta$ (interpreting $\mathrm{sgn}0=0$) gives $\mathrm{\parallel }(a,b){\mathrm{\parallel }}_{\mathrm{\infty }}\le 1$ and $\mathrm{\mid }\phi (a,b)\mathrm{\mid }=\mathrm{\mid }\alpha \mathrm{\mid }+\mathrm{\mid }\beta \mathrm{\mid }$, so $\mathrm{\parallel }\phi \mathrm{\parallel }=\mathrm{\mid }\alpha \mathrm{\mid }+\mathrm{\mid }\beta \mathrm{\mid }$. Applying this to the formula from step 2.1 yields $\mathrm{\parallel }{F}_c\mathrm{\parallel }=\frac{\mathrm{\mid }1+c\mathrm{\mid }+\mathrm{\mid }1-c\mathrm{\mid }}{2}\mathrm{.}$ This equals $1$ exactly when $-1\le c\le 1$.

Therefore every $c\in [-1,1]$ gives a norm-preserving extension of $f_0$. This computes explicitly the admissible interval predicted abstractly by [L1], and in particular is consistent with the existence statement of [L2].