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By Iranmanesh A., Khosravi B., Alavi S.H.

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Extra info for A Characterization of PSU3(q) for q>5

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For each point ~ e % , we obtain a nearest point we take the last point UYof its intersection with i. It is continuous since ~ is a Chehyshev set. is boundedly compact. The map- @(1) is the According to Schauder's principle, q has The latter must lie on ~% beyond ~; this contradicts the way ~ was chosen. This theorem was proved for the first time in [VI2]. 5. Chebyshev Sets in Metric Spaces Assume now that M is an arbitrary set in a metric space X . there is defined a set (possibly empty) ~(~)cX For each point ~ ~ X , , consisting of points in ~ nearest to M.

If the space ~ is smooth, the set ~ ~(~) is nonempty and acyclic, then ~ is boundedly compact, and if for all is convex. In connection with the questions of uniqueness and existence of approximations in the set M for elements ~ X the properties of the spaces X , equivalences have been established for a number of {hemselves; in addition, there have been established general- izations of the concept of a "sun," criteria for suns, criteria for Chebyshev sets, and also for sets M P(~) with various properties for the operation of projection or for the projections themselves.

Thenecessityof the condition is obvious: In ~ we specify an arbitrary density, rapidly decreasing at in- finity, say, ~ ( ~ ) = @-'m'~ , and we define ~ ( ~ K tributed over ~ K ~ ~ ~ as the center of gravity of the mass dis- with density ~(~). We verify the sufficiency of the condition for ~=~+{ . the existence of a continuous ~ the set ~ = ~0 , where ~o of ~ . This follows since, obviously, ~ C ~o follows. Let ~ 0 9 Every plane least one point of ~ . (Otherwise We can see that for ~ =~-I and is the interior of the convex hull and the inclusion ~o C ~ can be verified as P~(-~,q) containing ~5 ( ~ being its normal) contains at ~-| tating P (~,~) would separate two points ~, ~ ~ P (~,@) about ~ until it passes through ~ and then about Z up to the position ~ the point

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