On the full range of Zippin and inclusion indices of rearrangement-invariant spaces

Let X be a rearrangement-invariant space on [0, 1]. It is known that its Zippin indices \(\underline{\beta }{}_X,\overline{\beta }{}_X\) $\underline{\beta }{}_X,\overline{\beta }{}_X$ and its inclusion indices \(\gamma _X,\delta _X\) ${\gamma }_X,{\delta }_X$ are related as follows: \(0\le \underline{\beta }{}_X\le 1/\gamma _X \le 1/\delta _X\le \overline{\beta }{}_X\le 1\) $0\le \underline{\beta }{}_X\le 1/{\gamma }_X\le 1/{\delta }_X\le \overline{\beta }{}_X\le 1$ . We show that given \(\underline{\beta },\overline{\beta }\in [0,1]\) $\underline{\beta },\overline{\beta }\in [0,1]$ and \(\gamma ,\delta \in [1,\infty ]\) $\gamma ,\delta \in [1,\infty ]$ satisfying \(\underline{\beta }\le 1/\gamma \le 1/\delta \le \overline{\beta }\) $\underline{\beta }\le 1/\gamma \le 1/\delta \le \overline{\beta }$ , there exists a rearrangement-invariant space X such that \(\underline{\beta }{}_X=\underline{\beta }\) $\underline{\beta }{}_X=\underline{\beta }$ , \(\overline{\beta }{}_X=\overline{\beta }\) $\overline{\beta }{}_X=\overline{\beta }$ and \(\gamma _X=\gamma \) ${\gamma }_X=\gamma$ , \(\delta _X=\delta \) ${\delta }_X=\delta$ .