1
$\begingroup$

So I was reading a proof about the hyperplane separation theorem, and the proof uses this lemma, and I have trouble proving it.

Given a convex, open set $K$ in a normed space $X$ such that $0 \in K$, for each $x \in K$, the Minkowski functional is defined as $\rho_K(x) := \text{inf}\{t > 0: \frac{x}{t} \in K\}$. Since $K$ is open, $\exists r >0$ such that $B(0,r) \subseteq K$. The lemma states $\rho_K(x) \leq \|x\| / r$. I have been thinking of this lemma for a while, but still have no clue.

$\endgroup$
0
5
$\begingroup$

If $0<s<r$ and $y=\frac {sx} {||x||}$ then $||y||=s<r$ so $y \in B(0,r)$. Hence $y \in K$. By definition of $\rho_K$ this gives $\rho_K (x) \leq \frac {||x||} s$ (because $t=||x|| /s$ is in the set $\{t>0 ;x/t \in K$). Now take limit as $s$ increases to $r$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.