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 $K_n(a,t) \ge M_n(a,t)$ 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.

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.

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 $\lim_{n \to \infty} R_n(a_n,t) = 0$ ?

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 $K_n(a,t) \ge M_n(a,t)$ 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.