**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 $X$?
> 
> 
> 



The anti-symmetric property of the collection of all compact convex sets of a Banach space
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.)


