Abstract minimal Radon partition

One fascinating and very useful feature of minimal Radon partitions is that they can be obtained from the orientation function, i.e. on a purely combinatorial level, without considering coordinates. For this purpose we introduce abstract minimal Radon partitions for an abstract orientation function. It will turn out that a pair A, B is a minimal Radon partition of a conformation if and only if it is an abstract minimal Radon partition of the corresponding orientation function. Before giving a formal definition of abstract minimal Radon partition, we first consider an example ... 

We obtain an abstract minimal Radon partition as the pair A, B of sets of atoms where set A consists of all atoms a whose two signs agree, and set B consists of all atoms with opposite signs. In the exeunple, we have... 

Definition (Abstract minimal Radon partition) An abstract minimal Radon partition of an abstract orientation function over atoms 0,...,n I) is a pair (A.B) of disjoint subsets of the set of atoms,... 

Note that if x(cio> > i i> i+i> . 4) = 0, then a, is neither in A nor in B, i.e. an abstract minimal Radon partition A, B may contain fewer than five atoms. Still, as for non-abstract minimal Radon partitions, exactly one abstract minimal Radon partition is assigned to each set of five atoms. Now, considering a conformation and its orientation function (which is also an abstract orientation function), we can formulate the following... 

The implication of an abstract minimal Radon partition to minimal Radon partition is still true as long as the abstract minimal Radon partition consists of exactly five atoms. For abstract minimal Radon partitions with fewer than five atoms, a quadruple of atoms with assigned zero-orientation may not lie exactly in a plane. However, as the convex hulls nearly intersect , we can interprete this as some kind of tolerance for Radon partitions. For more details and a proof of Remark 4.16 refer to [105]. However, the relevance of the statement given there is not obvious at a first glance, so we note the following ... 

Oriented matroids are introduced in [25] on page 103 using the structure of (signed) circuits, here named abstract minimal Radon partitions. On pages 104f in [25] (example The CUBE ), it is demonstrated that minimal Radon partitions of a sequence of points correspond to the circuits of an oriented matroid. 

Theorem 3.5.5 in [25] finally states that the pairs of functions consisting of a chirotope X and its negative x are in one-to-one correspondence to oriented matroids. lemma 3.5.7 in [25] explains how to obtain the signed circuits (i.e. the abstract minimal Radon partitions) from the chirotope. See also Remark 3.7.3 in the same reference. 

Example (Figure 4.2c, cont.) We already mentioned that a central carbon atom and its four neighbors form a minimal Radon partition, such as in exeimple 4.2c. Thus, A, B = 0, (1,2,3,4 is a minimal Radon partition of conformation Ic. According to Remark 4.16, A, B is also an abstract minimal Radon partition of its orientation function. Knowing only the partial orientation function shown in the subsection on partial orientation functions, we have the foUowing situation, corresponding to the quintuple (0,1,2,3,4) ... 

Test 5. Ensure that x has no abstract minimal Radon partition of the form A, 0. ... 

Comparing the signs in the table, we obtain the abstract minimal Radon partition 0,1,2,3,4, 0. Thus the orientation function x is not realizable. 

A conformation cannot have a minimal Radon partition of the form A, 0, as the convex hull of the empty set is empty, and thus, trivially, its intersection with the convex hull of A cannot be nonempty. However, abstract PDFs may have abstract minimal Radon peutitions of the form A, 0. Such abstract PDFs are not realizable because of Remark 4.16. This reasoning is correct even for zero-tolerances. Thus we can restrict our candidate abstract POFs by the following test ... 

In Figure 4.9, an arrangement of six atoms with this orientation function is shown. Thus this function is realizable. However, since there are chemical bonds between 2-3, 3-4, and 0 5, this arrangement contains the chemically forbidden minimal Radon partition drawn in Figure 4.8a. Thus, this realization of x is not a chemical realization. Moreover, as any realization of the abstract POF has the minimal Radon partition... 

Here we applied the test to an abstract full orientation function and successfully excluded one that was not chemically realizable. The test is applicable only in situations where hve quadruples a,b, c, d), (a, b, c, e), a,b,d,e), ( , c, d,e) and (h.c.d.e) are either all selected or their function values can be obtained by known minimal Radon partitions. We can only determine the abstract Radon partition corresponding to the 5-set a, b, c, d, e in such cases and decide whether or not it is a forbidden one. However, if we select quadruples of type i) and ii), then no such cluster of five quadruples is available as a rule, and Test 5 and Test 6 do not exclude any candidate abstract POP. 

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