Введение в теорию игр (сторонняя методичка) (1184510), страница 47
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Ïîñêîëüêó K − êîìïàêò,áåç ïîòåðè îáùíîñòè ìîæíî ñ÷èòàòü, ÷òî ïîñëåäîâàòåëüíîñòè {y i (k)}ñõîäÿòñÿ. Íî lim δ(πk ) = 0. Ïîýòîìó ïðè k → ∞ äèàìåòð ñèìïëåêñàk→∞y 1 (k) · · · y n+1 (k) ñòðåìèòñÿ ê íóëþ. Ñëåäîâàòåëüíî, lim y i (k) = y ∗ , i =k→∞1, ..., n + 1. Ïîñêîëüêó ïðè âñåõ i y i (k) ∈ Ci , à ìíîæåñòâà Ci çàìêíóòû,n+1Tèìååì y ∗ ∈Ci .i=1265ÏðèëîæåíèåÓòî÷íèì ôîðìóëèðîâêó òåîðåìû 9.1. Ïóñòü Z − âûïóêëûé êîìïàêòâ E m è y 1 · · · y n+1 − ñîäåðæàùèéñÿ â íåì ñèìïëåêñ ìàêñèìàëüíîé ðàçìåðíîñòè.
Òîãäà ÷èñëî n ≤ m íàçûâàåòñÿ ðàçìåðíîñòüþ ìíîæåñòâà Z.Âûïóêëûé êîìïàêò Z ìàêñèìàëüíîé ðàçìåðíîñòè m èìååò âíóòðåííþþòî÷êó (íàïðèìåð, áàðèöåíòð ñèìïëåêñà y 1 · · · y m+1 ). Îáðàòíî, åñëè Z èìååò âíóòðåííþþ òî÷êó, òî åãî ðàçìåðíîñòü ìàêñèìàëüíà. Ïðåäïîëîæèì,÷òî Z èìååò ðàçìåðíîñòü n < m è ñîäåðæèò íóëü. Òîãäà ìîæíî ñ÷èòàòü,÷òî y n+1 = 0. Ïðè ýòîì Z ÿâëÿåòñÿ ïîäìíîæåñòâîì åâêëèäîâà ïðîñòðàíñòâà E n ñ áàçèñîì y 1 , ..., y n .Èç ñêàçàííîãî âûòåêàåò, ÷òî â òåîðåìå 9.1 áåç ïîòåðè îáùíîñòè ìîæíîïðåäïîëîæèòü, ÷òî âûïóêëûé êîìïàêò Z èìååò ìàêñèìàëüíóþ ðàçìåðíîñòü m.Äîêàçàòåëüñòâî òåîðåìû 9.1 (Áðàóýðà).
Ïóñòü K = x1 · · · xm+1 −m-ìåðíûé ñèìïëåêñ â E m . Ïî òåîðåìå Ï.2 (ñì. Ï3) íàéäåòñÿ ãîìåîìîðôèçì τ : Z → K. Îòîáðàæåíèå τ ◦ f ◦ τ −1 ïåðåâîäèò òî÷êó y ∈ K â òî÷êóτ (f (τ −1 (y))) ∈ K è ÿâëÿåòñÿ íåïðåðûâíûì. Åñëè y 0 ∈ K − íåïîäâèæíàÿòî÷êà îòîáðàæåíèÿ τ ◦ f ◦ τ −1 , òî x0 = τ −1 (y 0 ) − íåïîäâèæíàÿ òî÷êàîòîáðàæåíèÿ f : Z → Z. Ïîýòîìó áåç ïîòåðè îáùíîñòè ìîæíî ñ÷èòàòü,÷òî Z = K.ÏîëîæèìCi = {x ∈ K | λi (f (x)) ≤ λi (x)}, i = 1, ..., m + 1.Ïîñêîëüêó îòîáðàæåíèå f è ôóíêöèè λi (x) íåïðåðûâíû íà K , ìíîæåñòâà Ci çàìêíóòû.
Ïîêàæåì, ÷òî ìíîæåñòâà Ci óäîâëåòâîðÿþò óñëîâèþïðåäûäóùåé òåîðåìû. Ïóñòü x ∈ xj1 · · · xjs . Òîãäàj1jsλj (x) = 0 ∀j ∈/ {x , ..., x } ⇒sXλjr (x) = 1 ≥r=1sXλjr (f (x)).r=1Ñëåäîâàòåëüíî, íàéäåòñÿ íîìåð r, ïðè êîòîðîì λjr (x) ≥ λjr (f (x)) èx ∈ Cjr .
Ïî ïðåäûäóùåé òåîðåìå íàéäåòñÿ∗x ∈m+1\Ci ⇒ λi (x∗ ) ≥ λi (f (x∗ )), i = 1, ..., m + 1.i=1Íîm+1Xi=1∗λi (x ) =m+1Xλi (f (x∗ )) = 1.i=1266ÏðèëîæåíèåÏîýòîìóλi (x∗ ) = λi (f (x∗ )), i = 1, ..., m + 1 ⇒ x∗ = f (x∗ ).Äîêàçàòåëüñòâî òåîðåìû 12.1 (Êàêóòàíè). Êàê è ïðè äîêàçà-òåëüñòâå òåîðåìû Áðàóýðà, áåç ïîòåðè îáùíîñòè ìîæíî ñ÷èòàòü, ÷òî âûïóêëûé êîìïàêò Z èìååò ìàêñèìàëüíóþ ðàçìåðíîñòü m.Ïóñòü K = x1 · · · xm+1 − m-ìåðíûé ñèìïëåêñ â E m . Ïî òåîðåìå Ï.2(ñì.
Ï3) íàéäåòñÿ ãîìåîìîðôèçì ϕ : K → Z. Ðàññìîòðèì ïîñëåäîâàòåëüíîñòü {πk } ñèìïëèöèàëüíûõ ðàçáèåíèé ñèìïëåêñà K, äëÿ êîòîðîélim δ(πk ) = 0. Äëÿ ïðîèçêîëüíîãî k îïðåäåëèì ñëåäóþùåå íåïðåðûâíîåk→∞îòîáðàæåíèå Φk : K → Z. Äëÿ ëþáîé âåðøèíû x èç {πk } âûáåðåì òî÷êóΦk (x) ∈ Φ(ϕ(x)) è çàòåì ïðîäîëæèì îòîáðàæåíèå ëèíåéíî íà êàæäûé èçñèìïëåêñîâ, âõîäÿùèõ â {πk }, ò.å. åñëèx=m+1Xjλj x ,j=1m+1Xλj = 1, λj ≥ 0, j = 1, ..., m + 1,j=1− òî÷êà ñèìïëåêñà x1 · · · xm+1 ∈ {πk }, òî ïîëàãàåìkΦ (x) =m+1XΦk (xj ).j=1Î÷åâèäíî, ÷òî ïîñëåäíåå ðàâåíñòâî îïðåäåëÿåò íåïðåðûâíîå îòáðàæåíèå Φk : K → Z .
Òîãäà f k = ϕ−1 Φk : K → K ÿâëÿåòñÿ íåïðåðûâíûìîòîáðàæåíèåì è èìååò íåïîäâèæíóþ òî÷êó xk . Ïóñòü xk ïðèíàäëåæèòñèìïëåêñó x1k · · · xm+1k ∈ {πk }, ò.å.kx =m+1Xj=1λkj xjk ,m+1Xλkj = 1, λkj ≥ 0, j = 1, ..., m + 1.j=1Íå óìàëÿÿ îáùíîñòè, ìîæíî ñ÷èòàòü, ÷òî ïîñëåäîâàòåëüíîñòè{xk }, {xjk },{λkj },{Φk (xjk )}, j = 1, ..., m + 1, ñõîäÿòñÿ ïðè k → ∞. Ïîñêîëüêó äèàìåòðû ñèìïëåêñîâ x1k · · · xm+1k ñòðåìÿòñÿ ê íóëþ, ïîñëåäîâàòåëüíîñòè {xk } è {xjk } äîëæíû èìåòü îáùèé ïðåäåë x∗ ∈ K. Ïóñòülim λkj = λ∗j , lim Φk (xjk ) = η j , j = 1, ..., m + 1.k→∞k→∞267Èìååìϕ(xk ) = Φk (xk ) =m+1Xλkj Φk (xjk )(Π.2)j=1è lim ϕ(xjk ) = ϕ(x∗ ), lim ϕ(xk ) = ϕ(x∗ ) â ñèëó íåïðåðûâíîñòè îòáðàæåk→∞k→∞íèÿ ϕ. Ïåðåõîäÿ â ðàâåíñòâå (Π.2) ê ïðåäåëó ïðè k → ∞, ïîëó÷èìϕ(x∗ ) =m+1Xj=1λ∗j η j ,m+1Xλ∗j = 1, λ∗j ≥ 0, j = 1, ..., m + 1.j=1Èç çàìêíóòîñòè îòîáðàæåíèÿ Φ âûòåêàåò, ÷òîη j ∈ Φ(ϕ(x∗ )), j = 1, ..., m + 1.
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