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A. Juriˇsić in V. Batagelj: Verjetnostni račun in statistika 707 ✬ Ester Klein je leta 1933 predstavil naslednjo geometrijsko nalogo: Med petimi točkami v ravnini, od katerih nobene tri niso kolinearne (leˇzijo na premici), lahko vedno izberemo ˇstiri, ki določajo konveksen četverokotnik. Reˇsitev: Vpeljemo pojem konveksne ogrinjače ... Če je konveksna ogrinjača teh petih točk (a) petkotnik, potem vsake 4 točke med njimi sestavljajo konveksen četverokotnik, (b) ˇstirikotnik, potem so njegovi vrhovi tiste 4 točke, ki smo jih iskali, (c) trikotnik, potem ga lahko označimo z A, B in C, preostali točki pa z D in E, tako da sta točki A in B na isti s strani premice DE. V tem primeru je četverokotnik ABCD konveksen. ✫ Univerza v Ljubljani ▲ ❙ ▲ ▲ ● ❙ ▲ ▲ ☛ ✖ ▲ ✩ ✪
A. Juriˇsić in V. Batagelj: Verjetnostni račun in statistika 708 ✬ Nalogo lahko posploˇsimo na 9 točk in iskanje konveksnega petkotnika ter počasi pridemo do Erdöseve domneve, da za konveksen k-kotnik potrebujemo v ravnini vsaj ✫ n = 1 + 2 k−2 točk od katerih nobene 3 niso kolinearne. Pravzaprav se je najprej Szekeres prepričal, da za dovolj velik n vedno obstaja konveksen k-kotnik, potem pa je Erdös postavil svojo domnevo. Univerza v Ljubljani ▲ ❙ ▲ ▲ ● ❙ ▲ ▲ ☛ ✖ ▲ ✩ ✪
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✬ ✫ Ljubljana, 6. oktober 2008
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A. Juriˇsić in V. Batagelj: Verje
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