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In compact convexity a simplex with a closed extreme boundary is called a regular simplex or a Bauer simplex. Except for compacity, any Khinchin structure is a Bauer simplex. 5 shows that U behaves like a regular set for the boundary value problem of potential theory. The next result extends some commonly used properties of compact Bauer simplices. 9 Let Conditions A1)-A3) be satisfied. The following are equivalent: 1) U is a simplex. 6. of a function 3) E Affb(U) for any f E Convb(U). 4) Affb(U) is a Riesz space.

1 Let X and Y be completely regular spaces and let <1> be a map from Y into 1'(X). The triplet (X, Y, <1» is said to be a standard Khinchin space, with U as its associated Khinchin structure, if the following conditions are satisfied: 81) <1> is a homeomorphism from Y onto <1>(Y); 82) U = co( <1>(Y)) is a Khinchin structure on X; 83) oe(U) = <1>(Y). The standard Khinchin space is said to be of injective type if the Khinchin space (X,U) is of injective type. 2 Let (X, Y, <1» be a standard Khinchin space.

Clearly, injectivity of F is sufficient. On the other hand, if x, y E Xl and F(x) =1= F(y), we choose Xo = {a}, Uo = 0, ¢ : a f-t x, 'ljJ : a f-t y and obtain F 0 'ljJ = F 0 ¢, but ¢ =1= 'ljJ. 108]). 2): Now assume ¢ 0 F = 'ljJ 0 F. We have to prove that ¢ = 'ljJ. Since, for any y E X 2 , there exists a filter £ on Xl such that y = lime F, we have ¢(y) = lime ¢ 0 F = lime 'ljJ 0 F = 'ljJ(y) . 4 Let (Xl, UI ) be a Khinchin space, let X 2 be a completely regular space and F : Xl ~ X 2 a homeomorphism.