The comparison constants between $\lVert\cdot\rVert_1$, $\lVert\cdot\rVert_2$ and $\lVert\cdot\rVert_\infty$ on $\mathbb{R}^{2}$, and vectors attaining each

On $\mathbb{R}^{2}$ with the norms of The $p$-norms $\lVert x\rVert_p$ for rational $p \ge 1$, and $\lVert x\rVert_\infty$, the comparison chain of The finite and reverse triangle inequalities for a norm; and for $n \ge 1$ every norm $N$ on $\mathbb{R}^n$ satisfies $N(x) \le C\lVert x\rVert_1$ and is Lipschitz, hence continuous, for $d_2$ clause 3 reads

$\lVert x\rVert_\infty \;\le\; \lVert x\rVert_2 \;\le\; \lVert x\rVert_1 \;\le\; \iota(2)\,\lVert x\rVert_\infty, \qquad \lVert x\rVert_1 \;\le\; \sqrt{\iota(2)}\;\lVert x\rVert_2 .$

Each of these four constants is attained, so none can be improved:

• $e_0 = (1,0)$ has $\lVert e_0\rVert_\infty = \lVert e_0\rVert_2 = \lVert e_0\rVert_1 = 1$, so the first and second inequalities are equalities there;
• $(1,1)$ has $\lVert(1,1)\rVert_\infty = 1$, $\lVert(1,1)\rVert_2 = \sqrt{\iota(2)}$ and $\lVert(1,1)\rVert_1 = \iota(2)$, so the third and fourth inequalities are equalities there.

The general theorem For $n \ge 1$ all norms on $\mathbb{R}^n$ are equivalent supplies constants but no attaining vectors; that is what this computation adds.

Unit balls. Writing $B_p := \{\, x \in \mathbb{R}^{2} : \lVert x\rVert_p \le 1 \,\}$ for $p \in \{1,2,\infty\}$, the chain gives $B_1 \subseteq B_2 \subseteq B_\infty$, and both inclusions are strict: $(1,1)$ lies in $B_\infty$ and not in $B_2$, and $\bigl(3/\iota(5)\bigr)(1,1)$ lies in $B_2$ and not in $B_1$. The scalar has to be chosen strictly between $1/\iota(2)$ and $1/\sqrt{\iota(2)}$: at the endpoint $1/\iota(2)$ the vector $(1,1)/\iota(2)$ has $\lVert\cdot\rVert_1 = 1$ and so still lies in $B_1$.

• Sharpness is not the same as equivalence. For $n \ge 1$ all norms on $\mathbb{R}^n$ are equivalent asserts that constants exist and produces some; nothing in it says which are smallest. The computation above supplies attaining vectors, and those are what make the constants of the chain best possible on $\mathbb{R}^{2}$.

• Both attaining vectors are extreme in the expected way. A vector with a single nonzero coordinate makes all three norms agree; a vector whose two coordinates have equal absolute value spreads the mass as evenly as possible and is where $\lVert\cdot\rVert_1$ is largest relative to the other two. On $\mathbb{R}^{n}$ the same two vectors give equality with $\iota(n)$ and $\sqrt{\iota(n)}$ in place of $\iota(2)$ and $\sqrt{\iota(2)}$; only the case $n = 2$ is verified here.

• The strictness computation in step 2.3 is arithmetic, not geometry. The scalar $3/\iota(5)$ was chosen to lie strictly between $1/\iota(2)$ and $1/\sqrt{\iota(2)}$; any scalar in that open interval would serve, and the interval is nonempty exactly because $\sqrt{\iota(2)} < \iota(2)$.