By J. Diestel
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Additional info for Geometry of Banach spaces: Selected topics
Paj are defined in U with values in D, so some subsequence of them converges to a limit function -J; with the property -J;(O) = 0, and hence -J;(U) c D. p~/(t) = t that 'ljJ(-J;(t)) = t for all t E U. This implies that 'ljJ(D) :J Ea. paj (D) :J K for all sufficiently large j. paj -+ 'ljJ and 'ljJ is non-constant. paj (D) such that Aj -+ A and therefore Art 'ljJ(D). p is continuous. Now let U be any open subset of B n and take an arbitrary point z E U. Set a = z/Izl and consider the open subset Ua C D.
In other words, we must show that the preimage of E* under any injection T: JP* ~ JPn* is C-convex. If T- I (E*) is empty there is nothing to prove, so we assume that the range of T intersects E* . 12 we then have T-I(E*) = (tT(E))*. 6 we know that tT(E) is C-convex, and hence so is its (dual) complement (tT(E))*. In one dimension any set is linearly convex, and we now make the assumption that open or compact C-convex sets in JPn-1 are known to be linearly convex. Let E be an open or compact «::-convex set in JPn.
Now let E be an open C-convex set in lP 2 , and let T be the projection from a ~ E as before. We shall prove that the image T(E) is C-convex. In fact, since a continuous image of a non-empty connected set is always non-empty and connected, what we have to show is that the open set T(E) is a proper and simply connected subset of lP. Suppose that it is not a proper subset, and hence that all lines through a intersect E. This means that all intersections of the compact lP 2 \ E with lines passing through the point a are C-convex.