Definition Polar Functions for Non-Negative Convex Functions is Equivalent for Gauge Functions

Question

In Rockafellar's "Convex Analysis", the polar of a general non-negative convex function which vanishes at the origin is given as: $$ f^{\circ}(x^{*}) = \inf \{ \mu^{*} \geq 0 \mid \langle x, x^{*}\rangle\leq 1 + \mu^{*}f(x), \forall x \}.$$ This definition is followed up by the claim if $ f = k$ is a gauge (a non-negative positively homogenous convex function with $ k(0) = 0 $), then the definition reduces to the definition to that of the polar of a gauge: $$ k^{\circ} (x^{*}) = \inf\{ \mu^{*} \geq 0 \mid \langle x, x^{*}\rangle\leq \mu^{*}k(x), \forall x \}. $$ I can't seem to be able to show that these are equivalent for $ f = k $. Is there a trick I am missing for utilizing $ k $'s positively homogeneity to remove the "+1"?

Answer 1

I am also reading this book. I have a simple idea and don't know if it is correct. Since $f(\lambda x)=\lambda f(x),\forall \lambda>0$, we can get $\langle x,x^*\rangle\leqslant 1+\mu^*f(x)\Leftrightarrow \langle ax,x^*\rangle\leqslant 1+\mu^*f(ax)\Leftrightarrow \langle x,x^*\rangle\leqslant \frac{1}{a}+\mu^*f(x),\forall a>0,x.$ So, $\frac{1}{a}\rightarrow 0$.