For any positive number $t$, nonnegative integer $n$, and nonzero vector $a \in \mathbb R^n$, define


$$
\begin{split}
K\_n(a,t) &:= \inf\_{x \in \mathbb R^n} \|x-a\|\_2 + t\|x\|\_1,\\
M\_n(a,t) &:= \min(\|a\|\_2,t\|a\|\_1),\\
R\_n(a,t) &:= K\_n(a,t)/M\_n(a,t).
\end{split}
$$


**Note.** $K\_n$ defines the Peetre's K-functional between $(\mathbb R^n,\ell\_2)$ and $(\mathbb R^n,\ell\_1)$, where the $\ell\_p$ norm of a vector $x=(x^1,\ldots,x^n)$ is defined by $\|x\|\_p := (\sum\_i |x^i|^p)^{1/p}$.


**Question.** Is it possible to construct $a\_n \in \mathbb R^n$ for each $n$, such that $\lim\_{n \to \infty} R\_n(a\_n,t) = 0$ ?


Motivation
----------


Clearly, one always has $K\_n(a,t) \le M\_n(a,t)$ with equality when $n=1$. What is not clear is whether one can construct $a$ with growing dimension $n$ such that $K\_n(a,t) \ll M\_n(a,t)$ eventually.



Fix positive $t$. Construct $a_n \in \mathbb R^n$ such that $(\inf_x \|x-a_n\|_2 + t\|x\|_1 )/\min(\|a_n\|_2,t\|a_n\|_1) \to 0$
This is to extend Christian Remling's [comment](https://mathoverflow.net/questions/432168/fix-positive-t-construct-a-n-in-mathbb-rn-such-that-inf-x-x-a-n-2#comment1112378_432168) to all real $t>0$, with an explicit lower bound on $K/M$, where $K:=K\_n(a,t)$ and $M:=M\_n(a,t)$.


$\newcommand\norm[1]{\lVert#1\rVert}$The key here, as in Christian Remling's [comment](https://mathoverflow.net/questions/432168/fix-positive-t-construct-a-n-in-mathbb-rn-such-that-inf-x-x-a-n-2#comment1112378_432168), is the observation that $\norm x\_1\ge\norm x\_2$ for all $x\in\mathbb R^n$. Indeed, this observation implies
$$K
\ge\inf\_{x\in\mathbb R^n}\bigl(\lvert\norm x\_2-\norm a\_2\rvert+t\norm x\_2\bigr)
=\inf\_{u\ge0}\bigl(\lvert u-\norm a\_2|+tu\bigr)
=\min(1,t)\norm a\_2$$
and
$$M\le\norm a\_2,$$
whence
$$\frac KM\ge\min(1,t).$$
(In particular, $K\ge M$ if $t\ge1$.)


