A version of the BorsukUlam theorem states that a continuous antipodal map from the Msphere into euclidean Nspace has a zero provided that M is at least N. Clearly the general case follows from the case when M = N. But is the case when M >> N any easier to prove than the equidimensional case?
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I don't think so, since any antipodal (nonexistent) map $S^n\to S^{n1}$ would easily be "suspended" to an antipodal map $S^{n+1}\to S^n$. Iterating and composing would then yield antipodal maps $S^m\to S^{n1}$ with arbitrarily large $m$. The $n=2$ case is somewhat easier though. 

