This is the first in a series of papers concerned with Morse quasiflats, which are a generalization of Morse quasigeodesics to arbitrary dimension. In this paper we introduce a number of alternative definitions, and under appropriate assumptions on the ambient space we show that they are equivalent and quasi-isometry invariant; we also give a variety of examples. The second paper proves that Morse quasiflats are asymptotically conical and have canonically defined Tits boundaries; it also gives some first applications.
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