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Edit: According to comment of " Fedor Petrov", I revise my question

Are there two compact convex subsets $X,Y$ of a Banach space with the following property?

They are not homeomorphic spaces but $X$ can be embedded in $Y$ and $Y $ can be embedded in $X$?

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  • $\begingroup$ You mean infinite-dinensional Euclidean space? In finite dimension convex compact sets are topologically balls. $\endgroup$
    – Fedor Petrov
    Commented Jan 7, 2017 at 14:50
  • $\begingroup$ @FedorPetrov Thank you very much for your comment. I was not aware of this so I revise my question. $\endgroup$
    – Ali Taghavi
    Commented Jan 7, 2017 at 14:56

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If I understand you correctly, the question is answered in the book Bessaga-Pelczynski, "Selected topics in infinite-dimensional topology", Warsaw, 1975. Combine Theorem 3.1 and Proposition 3.1 in that book (on page 100). The answer is: all such infinite-dimensional sets are homeomorphic to the Hilbert cube. (As for finite-dimensional sets, the question was answered in the classical dimension theory.)

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