The Chiral Racemization Path Problem in -Dimensions and Mislows Label Paradox

THE CHIRAL RACEMIZATION PATH PROBLEM IN n-DIMENSIONS AND MISLOW S LABEL PARADOX 

These counterintuitive properties of racemization paths of three-dimensional labeled and unlabeled chiral tetrahedra, noted by Mislow, are referred to as Mislow s label paradox. More recently, it has been shown that Mislow s label paradox is general for n-chiral simplices in all finite dimensions n, and sufficient and necessary partial vertex labeling conditions have been given for chirality-preserving interconversion paths of mirror images of chiral -dimensional simplexes. 

Two relevant theorems were proven in an earlier study on n-dimensional transformations preserving or abandoning n-chirality. Here only the main results (Theorems 3 and 10 of the study ) are presented, using the explicit labeling notation for the proofs, see the original reference.  

A minimal relaxation of the unique labeling condition of statement (1) is sufficient for the existence of a chirality-preserving racemization path. Following the general proof given for the n-dimensional case, this is verified subsequently. 

Proof of statement (3). Point set S represents an n-chiral simplex in consequently, any subset of k vertices of 5 defines a unique k - 1)-dimensional hyperplane that contains these k points. Take the (n - 2)-dimensional, unique hyperplane Q that contains the first n - 1, uniquely labeled vertices, and take the unique (n - l)-dimensional hyperplane A that contains the first n vertices of 5. Hyperplane is a maximum achiral subset of 5. Furthermore, 

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