By João B. (Ed.) Prolla
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FE£ where ,.. f} {f stands for the set f}, {£ v {x E X v f (x) }. de.. This follows from the first remark following the definition of the relative boundary. Corollary 2 settles Example 3. It can be seen from Example 1 that one cannot expect to the equali ty Jf £ = Kor PC, £) in Proposi tion 3 without additional as- sumptions. Indeed, since a£x = X we have ample. However, we know from Chapter II that tion id 3 have is neither convex nor concave on Proposition 4 will make clear why ;/i£ '" JC ;/i () JC in this ex- £ since the func- [-1, + 1 J • Kor (JC,£)=£ and hence Furthermore "'£ JC * Kor (X,£) • The proof of Proposition 3 uses a property of the closure S of the Choquet boundary S a£x which holds for much smaller closed sets in certain cases.
2). Let us remark that, e: ... p. for each Ul P : = {U 1 x • •• x Un of neighbourhoods of 9. (I). Finally (3) U. ~ Rl x ••• x Rn. Ul. E ~ U in the basis (i=l, ... ,n)} Hence (2) follows from 4. (1) and is implied by 9. (2), very end of section 1. 53 (3) and by the remark at the 0 3. DISCUSSION OF THE MOTIVATING EXAMPLES In this final section, we will look at some of the known results in the case of our motivating examples of sheaves F (cf. 1 above) and will point out that, between some theorems in the literature, strong relations follow from our previous discussion.
1 above) and will point out that, between some theorems in the literature, strong relations follow from our previous discussion. ll the relevant articles, but we will rather illus- trate some of the ideas which might playa role, when one tries to apply the results of sections 1 and 2, by specific examples. n Frechet sheaf of holomorphic functions on a complex manifold X. :: 1) here. c. space sheaf of E-val ued holomorphic functions. When for short, A(R,E), H(R,E) instead of E, OE 0 on the prodis just F = 0, we wi 11 the write, AF(K,E), HF(K,E), respectively.