Một số kết quả về ánh xạ nón pháp tuyến của tập lồi đa diện có nhiều tuyến tính

❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷ ❚❘❺◆ ❑■▼ ❚❍❆◆❍ ▼❐❚ ❙➮ ❑➌❚ ◗❯❷ ❱➋ ⑩◆❍ ❳❸ ◆➶◆ P❍⑩P ❚❯❨➌◆ ❈Õ❆ ❚❾P ▲➬■ ✣❆ ❉■➏◆ ❈➶ ◆❍■➍❯ ❚❯❨➌◆ ❚➑◆❍ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❍➔ ◆ë✐ ✲ ✷✵✶✽ ❇❐ ●■⑩❖ ❉Ö❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷ ❚❘❺◆ ❑■▼ ❚❍❆◆❍ ▼❐❚ ❙➮ ❑➌❚ ◗❯❷ ❱➋ ⑩◆❍ ❳❸ ◆➶◆ P❍⑩P ❚❯❨➌◆ ❈Õ❆ ❚❾P ▲➬■ ✣❆ ❉■➏◆ ❈➶ ◆❍■➍❯ ❚❯❨➌◆ ❚➑◆❍ ❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ❣✐↔✐ t➼❝❤ ▼➣ sè✿ ✻✵ ✹✻ ✵✶ ✵✷ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ◆❣÷í✐ ữợ ✲ ✷✵✶✽ ▲❮■ ❈❷▼ ❒◆ ▲í✐ ✤➛✉ t✐➯♥ tỉ✐ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ ✈➔ s➙✉ s➢❝ t tợ ổ ữợ ữớ ữợ t t t ữợ ú ù tổ tr sốt q tr ✈➔ ❤♦➔♥ t❤✐➺♥ ❧✉➟♥ ✈➠♥ ♥➔②✳ ❚ỉ✐ ❝ơ♥❣ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ tỵ✐ P❤á♥❣ ❙❛✉ ✣↕✐ ❤å❝✱ ❝→❝ t❤➛② ❝æ ❣✐→♦ ❣✐↔♥❣ ❞↕② ❝❤✉②➯♥ ♥❣➔♥❤ ❚♦→♥ ●✐↔✐ t➼❝❤✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❍➔ ◆ë✐ ✷ ✤➣ ❣✐ó♣ ✤ï tỉ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ t↕✐ tr÷í♥❣✳ ◆❤➙♥ ❞à♣ ♥➔② tỉ✐ ❝ơ♥❣ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ✤➳♥ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ❜➧ ❧✉ỉ♥ ❝ê ✈ơ✱ ✤ë♥❣ ✈✐➯♥✱ ❣✐ó♣ ✤ï tỉ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ❤♦➔♥ t❤✐➺♥ ❧✉➟♥ ✈➠♥ ♥➔②✳ ❍➔ ◆ë✐✱ ♥❣➔② ✷✸ t❤→♥❣ ✶✵ ♥➠♠ ✷✵✶✽ ❚→❝ ❣✐↔ ❧✉➟♥ ✈➠♥ ❚r➛♥ ổ ữợ sỹ ữợ ổ ▼ët sè ❦➳t q✉↔ ✈➲ →♥❤ ①↕ ♥â♥ ♣❤→♣ t✉②➳♥ t ỗ õ t t ữủ ❤♦➔♥ t❤➔♥❤ ❜ð✐ sü ♥❤➟♥ t❤ù❝ ✈➔ t➻♠ ❤✐➸✉ ❝õ❛ ◆❣✉②➵♥ ❚❤à ❚♦➔♥✱ ❧✉➟♥ ✈➠♥ ❝❤✉②➯♥ ♥❣➔♥❤ t♦→♥ ❣✐↔✐ t➼❝❤ ✈ỵ✐ ✤➲ t➔✐✿ ✧ ❜↔♥ t❤➙♥ t→❝ ❣✐↔✳ ❚r♦♥❣ q✉→ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ✈➔ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥✱ t→❝ ❣✐↔ ✤➣ ❦➳ t❤ø❛ ♥❤ú♥❣ ❦➳t q✉↔ ❝õ❛ ❝→❝ ♥❤➔ ❦❤♦❛ ❤å❝ ✈ỵ✐ sü tr➙♥ trå♥❣ ✈➔ ❜✐➳t ì♥✳ ❍➔ ◆ë✐✱ ♥❣➔② ✷✸ t❤→♥❣ ✶✵ ♥➠♠ ✷✵✶✽ ❚→❝ ❣✐↔ ❧✉➟♥ ✈➠♥ ❚r➛♥ ❑✐♠ ❚❤❛♥❤ ▼ư❝ ❧ư❝ ▼Ð ✣❺❯ ❈❤÷ì♥❣ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✹ ✼ ✶✳✶ ❑❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ ❦❤æ♥❣ ❣✐❛♥ ✤è✐ ♥❣➝✉ ✈➔ ❦❤æ♥❣ ❣✐❛♥ ❆s♣❧✉♥❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỗ ❞✐➺♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✸ ◆â♥ ♣❤→♣ t✉②➳♥ ❋r➨❝❤❡t ✈➔ ♥â♥ ♣❤→♣ t✉②➳♥ ▼♦r❞✉❦❤♦✈✐❝❤ ❝õ❛ ♠ët t➟♣ ❤ñ♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✹ ✣è✐ ✤↕♦ ❤➔♠ ❋r➨❝❤❡t ✈➔ ✤è✐ ✤↕♦ ❤➔♠ ▼♦r❞✉❦❤♦✈✐❝❤ ✳ ✳ ✳ ❈❤÷ì♥❣ ✷ ◆â♥ ♣❤→♣ t t ỗ ✣è✐ ✤↕♦ ❤➔♠ ❋r➨❝❤❡t ❝õ❛ →♥❤ ①↕ ♥â♥ ♣❤→♣ t✉②➳♥ t ỗ õ t t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✷ ✣è✐ ✤↕♦ ❤➔♠ ▼♦r❞✉❦❤♦✈✐❝❤ õ t t ỗ ❞✐➺♥ ❝â ♥❤✐➵✉ t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❑➳t ❧✉➟♥ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✷✽ ✹✸ ✹✹ ✸ ▲í✐ ♥â✐ ✤➛✉ ✶✳ ỵ t ố ữ õ ởt tr♦♥❣ ♥❤ú♥❣ ❧➽♥❤ ✈ü❝ ❦✐♥❤ ✤✐➸♥ ❝õ❛ t♦→♥ ❤å❝ ❝â ↔♥❤ ❤÷ð♥❣ ✤➳♥ ❤➛✉ ❤➳t ❝→❝ ❧➽♥❤ ✈ü❝ ❦❤♦❛ ❤å❝ ✲ ❝æ♥❣ ♥❣❤➺ ✈➔ ❦✐♥❤ t➳ ✲ ①➣ ❤ë✐✳ ❚r♦♥❣ t❤ü❝ t➳✱ ✈✐➺❝ t➻♠ ❣✐↔✐ ♣❤→♣ tè✐ ÷✉ ❝❤♦ ♠ët ✈➜♥ ✤➲ ♥➔♦ ✤â ❝❤✐➳♠ ♠ët ✈❛✐ trá ❤➳t sù❝ q✉❛♥ trå♥❣✳ P❤÷ì♥❣ →♥ tè✐ ÷✉ ❧➔ ♣❤÷ì♥❣ →♥ ❤đ♣ ỵ t tốt t tt t ỗ ỹ q t tố ữ ỡ tr ỵ tt tố ữ t t ỹ t ởt số ữợ ♠ët sè r➔♥❣ ❜✉ë❝✳ ❇➔✐ t♦→♥ tè✐ ÷✉ ❝â ♠è✐ q✉❛♥ ❤➺ ♠➟t t❤✐➳t ✈ỵ✐ ♠ët sè ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥ ✤➳♥ tè✐ ÷✉✿ tø ❜➜t ✤➥♥❣ t❤ù❝ ❑② ❋❛♥ ỏ ữủ t ợ t tổ ỡ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣✮✱ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥✱ ❜➔✐ t♦→♥ ✤✐➸♠ ②➯♥ ♥❣ü❛✱ ❜➔✐ t♦→♥ ❜ò✱✳✳✳ ✤➳♥ ❝→❝ ❜➔✐ t♦→♥ r➜t t❤ü❝ t✐➵♥ ❧➔ trá ❝❤ì✐ ❦❤ỉ♥❣ ❤đ♣ t→❝ ✭❝ô♥❣ ❣å✐ ❧➔ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ ◆❛s❤✮✱ ❜➔✐ t♦→♥ ♠↕♥❣ ❣✐❛♦ t❤ỉ♥❣ ✈➔ ♥➲♥ ❦✐♥❤ t➳ t❤✉➛♥ tó② tr❛♦ ✤ê✐✳ ❑❤→✐ ♥✐➺♠ →♥❤ ①↕ ✤❛ trà ①✉➜t ❤✐➺♥ tø ♥❤ú♥❣ ♥➠♠ ✸✵ ❝õ❛ t❤➳ ❦✛ ✷✵ tr➯♥ ❝ì sð ♥❤ú♥❣ ❜➔✐ t♦→♥ ❝â tr♦♥❣ t❤ü❝ t➳✳ ❈→❝ ❜➔✐ t♦→♥ tố ữ tr ợ t tứ t❤➟♣ ♥✐➯♥ ✽✵ ❝õ❛ t❤➳ ❦✛ ✷✵✱ ♠ð ✤➛✉ ❜ð✐ ❝→❝ ❝æ♥❣ tr➻♥❤ ❝õ❛ ❏✳ ▼✳ ❇♦r✇❡✐♥ ♥➠♠ ✶✾✽✶✱ ❱✳ P♦st♦❧✐❝➠ ♥➠♠ ✶✾✽✻ ✈➔ ❍✳ ❲✳ ❈♦r❧❡② ♥➠♠ ✶✾✽✼ ♥❤÷♥❣ ✤➣ ♥❤➟♥ ✤÷đ❝ ♥❤✐➲✉ sü q✉❛♥ t➙♠ ❝õ❛ ❝→❝ ♥❤➔ t♦→♥ ❤å❝ ✈➔ ①✉➜t ❤✐➺♥ ♥❣➔② ❝➔♥❣ ♥❤✐➲✉ tr➯♥ ❝→❝ t t tr ỵ t❤✉②➳t tè✐ ÷✉ ❝ơ♥❣ ❞➛♥ ❞➛♥ ✤÷đ❝ ♠ð rë♥❣ ❝❤♦ →♥❤ ①↕ ✤❛ trà ✈➔ ❤➻♥❤ t❤➔♥❤ ♥➯♥ ♠ët ♥❣➔♥❤ t õ ỵ tt tố ÷✉ ✤❛ trà✳ ✣è✐ ✤↕♦ ❤➔♠ ❝õ❛ ♠ët →♥❤ ①↕ ✭✤è✐ ✤↕♦ ❤➔♠ ❋r➨❝❤❡t✱ ✤è✐ ✤↕♦ ❤➔♠ ▼♦r❞✉❦❤♦✈✐❝❤✱✳✳✳✮ ❧➔ ❝→❝ ❦❤→✐ ♥✐➺♠ ❦❤æ♥❣ t❤➸ t❤✐➳✉ tr♦♥❣ ❣✐↔✐ t➼❝❤ ✤❛ trà✳ ✣➦❝ ❜✐➺t✱ ✤è✐ ✤↕♦ ❤➔♠ ❝õ❛ ♠ët →♥❤ ①↕ t✉②➳♥ t tữỡ ự ợ t tỷ ủ ♥â✳ ❇➯♥ ❝↕♥❤ ✤â✱ ♥❤÷ ❝❤ó♥❣ t❛ ✤➣ ❜✐➳t✱ ❝→❝❤ t✐➳♣ ❝➟♥ ❦❤æ♥❣ ❣✐❛♥ ✤è✐ ♥❣➝✉ ♥❤✐➲✉ ❦❤✐ r➜t ❤ú✉ ❤✐➺✉✱ ❝â ♥❤ú♥❣ tr÷í♥❣ ❤đ♣ ❝á♥ ❤ú✉ ❤✐➺✉ ❤ì♥ ❝↔ ❝→❝❤ t✐➳♣ ❝➟♥ ❜➡♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ♥➲♥✳ ◆❤÷ ✈➟②✱ ❜➯♥ ❝↕♥❤ ❝→❝ t➼♥❤ ❝❤➜t ❧✐➯♥ tö❝ ❝õ❛ →♥❤ ①↕ ✤❛ trà ✭t➼♥❤ ♥û❛ ❧✐➯♥ tö❝ tr➯♥✱ t➼♥❤ ♥û❛ ❧✐➯♥ tö❝ ữợ t st t st →♥❤ ①↕ ✭t❤❡♦ ♥❣❤➽❛ ❝ê ✤✐➸♥ ❤♦➦❝ t❤❡♦ ♥❣❤➽❛ s✉② rë♥❣✮✱ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ✤è✐ ✤↕♦ ❤➔♠ ❝õ❛ ♠ët →♥❤ ①↕ ✤❛ trà ❧➔ ♠ët ✈➜♥ ✤➲ ❝ì ❜↔♥ tr♦♥❣ ❣✐↔✐ t➼❝❤ ✤❛ trà✳ ✣➣ ❝â ♥❤✐➲✉ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ✤è✐ ✤↕♦ ❤➔♠ ❝õ❛ ♠ët →♥❤ ①↕ ✤❛ trà ①❡♠ ❬✶✲✾❪ ✈➔ ♥❤ú♥❣ t➔✐ ❧✐➺✉ tr➼❝❤ ❞➝♥ tr♦♥❣ ✤â✳ ✧▼ët sè ❦➳t q✉↔ ✈➲ →♥❤ ①↕ ♥â♥ ♣❤→♣ t t ỗ õ t t ❝❤♦ ❧✉➟♥ ✈➠♥ ♥➔②✳ ❱ỵ✐ ♥❤ú♥❣ ❧➼ ❞♦ tr➯♥✱ ❝❤ó♥❣ tæ✐ ❝❤å♥ ✤➲ t➔✐ ♥❣❤✐➯♥ ❝ù✉ ◆❣♦➔✐ ♣❤➛♥ ▼ð ✤➛✉✱ ❑➳t ❧✉➟♥ ✈➔ ❉❛♥❤ ♠ö❝ ❝→❝ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✱ trú ỗ ữỡ ữ s ❈❤÷ì♥❣ ✶ ✧❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à✧ tr➻♥❤ ❜➔② ✈➲ ♠ët sè ❦❤→✐ ♥✐➺♠ ✈➲ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ ❦❤æ♥❣ ❣✐❛♥ ✤è✐ ổ s t ỗ ố ❤➔♠ ❋r➨❝❤❡t ✈➔ ✤è✐ ✤↕♦ ❤➔♠ ▼♦r❞✉❦❤♦✈✐❝❤✳ ❈❤÷ì♥❣ ✷ ✧◆â♥ t t ỗ tr ✤à♥❤ ❧➼✱ ♠➺♥❤ ✤➲✱ ❜ê ✤➲ ✈➲ ✤è✐ ✤↕♦ ❤➔♠ ❋r➨❝❤❡t ✈➔ ✤è✐ ✤↕♦ ❤➔♠ ▼♦r❞✉❦❤♦✈✐❝❤✳ ✷✳ ▼ö❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉ ▼ët ♠➦t✱ ✤÷❛ r❛ ❝ỉ♥❣ t❤ù❝ ❝❤♦ ♥â♥ ♣❤→♣ t t ỗ õ t t ▼➦t ❦❤→❝✱ t❤✐➳t ❧➟♣ ❝→❝ ❝æ♥❣ t❤ù❝ t➼♥❤ t♦→♥ ✤è✐ ✤↕♦ ❤➔♠ ❋r➨❝❤❡t ✈➔ ✤è✐ ✤↕♦ ❤➔♠ ▼♦r❞✉❦❤♦✈✐❝❤ ❝❤♦ →♥❤ ①↕ ♥â♥ ♣❤→♣ t✉②➳♥✳ ✸✳ ◆❤✐➺♠ ✈ư ♥❣❤✐➯♥ ❝ù✉ ✣÷❛ r ổ tự ố rt ữợ ữủ tr ữợ ố ✤↕♦ ❤➔♠ ▼♦r❞✉❦❤♦✈✐❝❤ ❝õ❛ →♥❤ ①↕ ♥â♥ ♣❤→♣ t✉②➳♥ ❝õ❛ t ỗ õ t t ✣è✐ t÷đ♥❣ ✈➔ ♣❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉ ▲✉➟♥ ✈➠♥ t➟♣ tr✉♥❣ ♥❣❤✐➯♥ ❝ù✉ ❝æ♥❣ t❤ù❝ ❝❤♦ ♥â♥ ♣❤→♣ t✉②➳♥ ❝õ❛ t ỗ õ t t ✤è✐ ✤↕♦ ❤➔♠ ❝õ❛ →♥❤ ①↕ ♥â♥ ♣❤→♣ t✉②➳♥ ✤â✳ ✺✳ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉ ❙û ❞ư♥❣ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉ tr♦♥❣ ❣✐↔✐ t➼❝❤ ❜✐➳♥ ♣❤➙♥ ✈➔ ✤↕♦ ❤➔♠ s✉② rë♥❣✱ ✤↕✐ sè t✉②➳♥ t➼♥❤✱ ❣✐↔✐ t➼❝❤ ✤❛ trà✱ t ỗ ỵ tt tố ữ õ ❣â♣ ❝õ❛ ❧✉➟♥ ✈➠♥ ❚r➻♥❤ ❜➔② ❝â ❤➺ t❤è♥❣ ✈➔ t÷ì♥❣ ✤è✐ ✤➛② ✤õ ✈➲ →♥❤ ①↕ ♥â♥ ♣❤→♣ t✉②➳♥ t ỗ õ t t õ t❤➸ sû ❞ö♥❣ ❧➔♠ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❝❤♦ s✐♥❤ ✈✐➯♥✱ ❤å❝ ✈✐➯♥ ❝❛♦ ❤å❝ ❝â q✉❛♥ t➙♠ ✤➳♥ ❧➽♥❤ ✈ü❝ t♦→♥ ❣✐↔✐ t➼❝❤✱ ❣✐↔✐ t➼❝❤ ✤❛ trà✳ ✻ ❈❤÷ì♥❣ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ❝õ❛ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ♥❤÷ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝✱ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✱ ❦❤æ♥❣ ❣✐❛♥ ✤è✐ ♥❣➝✉✱ ❦❤æ♥❣ ❣✐❛♥ tæ♣æ✱ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ ❦❤æ♥❣ ❣✐❛♥ ❆s♣❧✉♥❞✱ õ t t ỗ ❝â ♥❤✐➵✉✱ ♥â♥ ♣❤→♣ t✉②➳♥ ❋r➨❝❤❡t✱ ♥â♥ ♣❤→♣ t✉②➳♥ ▼♦r❞✉❦❤♦✈✐❝❤ ❝õ❛ ♠ët t➟♣ ❤ñ♣✱ ✤è✐ ✤↕♦ ❤➔♠ ❝õ❛ ♠ët →♥❤ ①↕ ✤❛ trà ✭✤è✐ ✤↕♦ ❤➔♠ ❋r➨❝❤❡t✱ ✤è✐ ✤↕♦ ❤➔♠ ▼♦r❞✉❦❤♦✈✐❝❤✮✳ ✶✳✶ ❑❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ ❦❤æ♥❣ ❣✐❛♥ ✤è✐ ♥❣➝✉ ✈➔ ❦❤ỉ♥❣ ❣✐❛♥ ❆s♣❧✉♥❞ ▼ư❝ ♥➔② tr➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ❝õ❛ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ♥❤÷ ❦❤ỉ♥❣ ❣✐❛♥ ♠❡tr✐❝✱ ❦❤ỉ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✱ ❦❤æ♥❣ ❣✐❛♥ ✤è✐ ♥❣➝✉✱ ❦❤æ♥❣ ❣✐❛♥ tæ♣æ✱ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ ❦❤æ♥❣ ❣✐❛♥ ❆s♣❧✉♥❞✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ ❚❛ ❣å✐ ❦❤ỉ♥❣ ❣✐❛♥ ♠❡tr✐❝ ❧➔ ♠ët t➟♣ ❤đ♣ X = ∅ ❝ò♥❣ ✈ỵ✐ ♠ët →♥❤ ①↕ d tø t➼❝❤ ❉❡s❝❛rt❡s X × X ✈➔♦ t➟♣ ❤đ♣ sè t❤ü❝ R t❤ä❛ ♠➣♥ ❝→❝ t✐➯♥ ✤➲ s❛✉ ✤➙②✿ ✶✳ d(x, y) ∀x, y ∈ X; d(x, y) = ⇔ x = y t ỗ t d(x, y) = d(y, x) ∀x, y ∈ X ✭t✐➯♥ ✤➲ ✤è✐ ①ù♥❣✮✳ ✸✳ d(x, y) d(x, z) + d(z, y) ∀x, y, z ∈ X ✭t✐➯♥ ✤➲ t❛♠ ❣✐→❝✮✳ ✼ ⑩♥❤ ①↕ d ✤÷đ❝ ❣å✐ ❧➔ ♠❡tr✐❝ tr➯♥ X, sè d(x, y) ❣å✐ ❧➔ ❦❤♦↔♥❣ ❝→❝❤ ❣✐ú❛ ❤❛✐ ♣❤➛♥ tû x ✈➔ y ❈→❝ ♣❤➛♥ tû ❝õ❛ X ❣å✐ ❧➔ ❝→❝ ✤✐➸♠❀ ❝→❝ t✐➯♥ ✤➲ ✶✮✱ ✷✮✱ ✸✮ ❣å✐ ❧➔ ❤➺ t✐➯♥ ✤➲ ♠❡tr✐❝✳ ❑❤ỉ♥❣ ❣✐❛♥ ♠❡tr✐❝ ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ M = (X, d) ❱➼ ❞ư ✶✳✶✳ ❚➟♣ ❤đ♣ ❝→❝ sè t❤ü❝ R ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ♠❡tr✐❝✱ ✈ỵ✐ ♠❡tr✐❝ d(x, y) = |x − y|, x, y ∈ R ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳ ❈❤♦ M = (X, d)✳ ❉➣② ✤✐➸♠ (xn) ⊂ X ❣å✐ ❧➔ ❞➣② ❝ì ❜↔♥ ✭❤❛② ❞➣② ❈❛✉❝❤②✮ tr♦♥❣ M ✱ ♥➳✉ ∀ε > 0, ∃n0 ∈ N ∗ s❛♦ ❝❤♦ ∀m, n t❛ ❝â d(xn , xm ) < ε ❤❛② n0 , lim d(xn , xm ) = n,m→∞ ❉➵ ❞➔♥❣ t❤➜② ♠å✐ ❞➣② ✤✐➸♠ (xn ) ⊂ X ❤ë✐ tö tr♦♥❣ M ✤➲✉ ❧➔ ❞➣② ❝ì ❜↔♥✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✸✳ ❑❤ỉ♥❣ ❣✐❛♥ ♠❡tr✐❝ M = (X, d) ❣å✐ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✤➛② ♥➳✉ ♠å✐ ❞➣② ❝ì ❜↔♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ♥➔② ❤ë✐ tư✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✹✳ ❑❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ ✭❤❛② ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ✤à♥❤ ❝❤✉➞♥✮ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ X tr➯♥ tr÷í♥❣ K (K = R ❤♦➦❝ K = C) ❝ò♥❣ ✈ỵ✐ ♠ët →♥❤ ①↕ tø X ✈➔♦ t➟♣ sè t❤ü❝ R✱ ❦➼ ❤✐➺✉ ❧➔ ✈➔ ✤å❝ ❧➔ ❝❤✉➞♥✱ t❤ä❛ ♠➣♥ ❝→❝ t✐➯♥ ✤➲ s❛✉ ✤➙②✿ ✶✮ x ∀x ∈ X; x = ⇐⇒ x = θ ✭❦➼ ❤✐➺✉ ♣❤➛♥ tû ❦❤æ♥❣ ❧➔ θ✮✳ ✷✮ αx = |α| x ✸✮ x + y ∀x ∈ X, ∀α ∈ K x + y ∀x, y ∈ X ❙è x ❣å✐ ❧➔ ❝❤✉➞♥ ❝õ❛ ✈❡❝t♦ x✳ ❚❛ ❝ơ♥❣ ❦➼ ❤✐➺✉ ❦❤ỉ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ ❧➔ X ✳ ❈→❝ t✐➯♥ ✤➲ 1), 2), 3) ❣å✐ ❧➔ ❤➺ t✐➯♥ ✤➲ ❝❤✉➞♥✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✺✳ ❑❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ X ❣å✐ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ♥➳✉ ♠å✐ ❞➣② ❝ì ❜↔♥ tr♦♥❣ X ✤➲✉ ❤ë✐ tö✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✻✳ ❈❤♦ t➟♣ X ❜➜t ❦ý✳ ❚❛ ♥â✐ ♠ët ❤å τ ♥❤ú♥❣ t➟♣ ❝♦♥ ❝õ❛ X ❧➔ ♠ët tæ♣æ tr➯♥ X ✱ ♥➳✉ ♥â t❤ä❛ ♠➣♥ ❝→❝ t➼♥❤ ❝❤➜t s❛✉✿ ✽ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❝❤✐➲✉ ♥❣÷đ❝ ❧↕✐ BI,K ⊃ T x¯; Θ(¯b) ∩ {¯ x∗ }⊥ ❑❤✐ ✤â T x¯; Θ(¯b) ∩ {¯ x∗ }⊥ = BI,K ❑➳t ❤đ♣ ✈ỵ✐ ❇ê ✤➲ ✷✳✷✱ t❛ t❤✉ ✤÷đ❝ T x¯; Θ(¯b) ∩ {¯ x∗ }⊥ ∗ = (BI,K )∗ = AI,K ❚❛ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❇ê ✤➲ tr➯♥✳ ❇ê ✤➲ ✷✳✺✳ ✭❳❡♠ ❬✽✱ ❇ê ✤➲ ✹✳✹❪✮ ❈❤♦ (¯x, ¯b, x¯∗) ∈ gph F, I = I(¯x, ¯b), λ = (λi )i∈I ✈ỵ✐ K := {i ∈ I : λi > 0}✳ ❑❤✐ ✤â✱ N (¯ x, ¯b, x¯∗ ); gph F = (x∗ , b∗ , v) : (x∗ , v) ∈ AI,K × BI,K , b∗i a∗i , b∗I¯ = 0, b∗I1 x∗ = − , ✭✷✳✷✶✮ i∈I ✈ỵ✐ I¯ = T \I ✈➔ I1 = I1 (¯ x, ¯b, x¯∗ ) ❱ỵ✐ (x, b, x∗ ) ∈ gph F, t❛ ✤➦t I(x, b, x∗ ) = {P ⊂ I(x, b) : P = ∅, x∗ ∈ ♣♦s{a∗i : i ∈ P }}, J (x, b, x∗ ) = {P ∈ I(x, b, x∗ ) : a∗i , i ∈ P, ❧➔ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤}, ✈➔ I(x, b, x∗ ) = J (x, b, x∗ ) ♥➳✉ x∗ = 0, J (x, b, x∗ ) ∪ {∅} ♥➳✉ x∗ = ❱ỵ✐ Q ⊂ T ✱ t❛ ✤à♥❤ ♥❣❤➽❛ FQ (b) = {x ∈ X : a∗i , x = bi ∀i ∈ Q, a∗i , x < bi ∀i ∈ T \Q} ❈❤♦ (¯ x, ¯b, x¯∗ ) ∈ gph F, I = I(¯ x, ¯b), J = I\I1 (¯ x, ¯b, x¯∗ ), I = I(¯ x, ¯b, x¯∗ )✱ ✸✶ ✈➔ I = I(¯ x, ¯b, x¯∗ ) ✣à♥❤ ♥❣❤➽❛ Σ(¯ x, ¯b, x¯∗ ) = (x∗ , b∗ , v) : (x∗ , v) ∈ AQ,P × BQ,P , P ⊂Q⊂I, P ∈I x∗ = − b∗i a∗i , b∗Q = 0, b∗Q\P ✭✷✳✷✷✮ i∈Q ✈➔ Σ0 (¯ x, ¯b, x¯∗ ) = (x∗ , b∗ , v) : (x∗ , v) ∈ AQ,P × BQ,P , P ⊂Q⊂I, P ∈I FQ (¯b)=∅ ∗ b∗i a∗i , bQ = 0, b∗Q\J x∗ = − ✭✷✳✷✸✮ i∈Q ❚❛ ❝❤➾ r❛ r➡♥❣ ❝→❝ t➟♣ ❤ñ♣ tr♦♥❣ ✭✷✳✷✷✮ ✈➔ ✭✷✳✷✸✮ ❧➛♥ ❧÷đt ❧➔ ✤→♥❤ ❣✐→ tr➯♥ ✈➔ ✤→♥❤ ữợ ợ õ t N ( x, ¯b, x¯∗ ); gph F ✳ ✣à♥❤ ❧➼ ✷✳✹✳ ✭❳❡♠ ❬✽✱ ✣à♥❤ ❧➼ ✹✳✶❪✮ ❈❤♦ (¯x, ¯b, x¯∗) ∈ gph F õ t õ ữợ ữủ s Σ0 (¯ x, ¯b, x¯∗ ) ⊂ N (¯ x, ¯b, x¯∗ ); gph F ⊂ Σ(¯ x, ¯b, x¯∗ ), ✭✷✳✷✹✮ ✈ỵ✐ Σ(¯ x, ¯b, x¯∗ ) ✈➔ Σ0 (¯ x, ¯b, x¯∗ ) ❧➛♥ ❧÷đt ❝❤♦ ❜ð✐ ✭✷✳✷✷✮ ✈➔ ✭✷✳✷✸✮✳ ❍ì♥ ♥ú❛✱ ♥➳✉ x¯∗ = 0✱ t❤➻ N (¯ x, ¯b, x¯∗ ); gph F ⊂ Σ0 (¯ x, ¯b, x¯∗ ) ⊂ N (¯ x, ¯b, x¯∗ ); gph F ự ú ỵ r gph F ❧➔ ✤â♥❣ ❜ð✐ ❇ê ✤➲ ✶✳✷ ✈➔ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♣❤↔♥ ①↕✳ ❉♦ ✤â✱ t❤❡♦ ❬✹✱ ✣à♥❤ ❧➼ ✷✳✸✺❪✱ N (¯ x, ¯b, x¯∗ ); gph F = ▲✐♠ sup N (xk , bk , x∗k ); gph F gph F (xk ,bk ,x∗k )− −−→(¯x,¯b,¯x∗ ) ✣➸ t❤✉ ✤÷đ❝ ❜❛♦ ❤➔♠ t❤ù ❤❛✐ tr♦♥❣ ✭✷✳✷✹✮✱ t❛ ❝è ✤à♥❤ (x∗ , b∗ , v) ∈ N (¯ x, ¯b, x¯∗ ); gph F ✸✷ ❚❛ s➩ ❝❤➾ r❛ r➡♥❣ (x∗ , b∗ , v) ∈ Σ(¯ x, ¯b, x¯∗ )✳ ◗✉❛ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ ❣✐ỵ✐ ❤↕♥ gph F w∗ ♥â♥ ♣❤→♣ t✉②➳♥✱ ❝â t❤➸ t➻♠ (xk , bk , x∗ ) −−−→ (¯ x, ¯b, x¯∗ ) ✈➔ (u∗ , η ∗ , vk ) −→ k (x , b , v) s❛♦ ❝❤♦ ∗ k k ∗ (u∗k , ηk∗ , vk ) ∈ N (xk , bk , x∗k ); gph F ✈ỵ✐ ♠å✐ k ∈ N✳ ❚ø I(xk , bk ) ⊂ T ✈➔ (xk , bk ) → (¯ x, ¯b) ✈ỵ✐ ♠å✐ k ✱ t❛ ❝â t❤➸ ❣✐↔ sû r➡♥❣ I(xk , bk ) = Q ✈ỵ✐ ♠å✐ k ✱ tr♦♥❣ ✤â Q ⊂ I(¯ x, ¯b) ❧➔ t➟♣ ❝❤➾ sè ①→❝ ✤à♥❤✳ ❉♦ ✤â✱ t❤❡♦ ▼➺♥❤ ✤➲ ✶✳✶✱ x∗k ∈ N xk ; Θ(bk ) = ♣♦s{a∗i : i ∈ Q} ✈ỵ✐ ♠å✐ k ∈ N ❚❤❡♦ ❇ê ✤➲ ✶✳✶ ✈➔ ◆❣✉②➯♥ ❧➼ ❉✐r✐❝❤❧❡t✱ ①➨t ❞➣② ❝♦♥ ❝õ❛ {x∗k } ♥➳✉ ❝➛♥ t❤✐➳t✱ t❛ ❝â t❤➸ ❣✐↔ t❤✐➳t r➡♥❣ tỗ t ởt t P Q s ✈❡❝t♦ {a∗i : i ∈ P } ❧➔ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ ✈➔ x∗k ∈ ♣♦s{a∗i : i ∈ P } ✈ỵ✐ ♠å✐ k ∈ N ❈â ♥❣❤➽❛ ❧➔ λki a∗i x∗k = ✈ỵ✐ λki 0; i ∈ P i∈P ❙û ❞ö♥❣ ◆❣✉②➯♥ ❧➼ ❉✐r✐❝❤❧❡t✱ t❛ ❝â t❤➸ t➻♠ ❞➣② ❝♦♥ {k } ❝õ❛ {k} ✈➔ ♠ët t➟♣ P ⊂ P s❛♦ ❝❤♦ {i ∈ P : λki > 0} = P, ❱ỵ✐ ✈ỵ✐ ♠å✐ ∈ N ✭✷✳✷✻✮ ∈ N✱ (u∗k , ηk∗ , vk ) ∈ N ((xk , bk , x∗k ); gph F)✱ t❤❡♦ ❇ê ✤➲ ✷✳✺✱ t❛ ❝â (u∗k , vk ) ∈ AQ,P × BQ,P , u∗k = − (ηk∗l )i a∗i , (ηk∗ )Q¯ = 0, i∈Q ∗ (ηk )I1 (xk ,bk ,x∗k ) ✭✷✳✷✼✮ ◆➳✉ P = ∅, t❤➻ x∗k = ợ N r trữớ ủ t ❝â x¯∗ = ✈➻ x¯∗ = lim →∞ x∗k ✳ ❚❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ I1 (xk , bk , x∗k )✱ t❛ t❤➜② ✸✸ I1 (xk , bk , x∗k ) = Q, ✈➔ Q\P = I1 (xk , bk , x∗k ) ◆➳✉ P = ∅, t❤➻ x∗k = λki a∗i ✈ỵ✐ ♠å✐ ✭✷✳✷✽✮ ∈ N i∈P ❱➻ {a∗i : i ∈ P } ❧➔ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ t❤❡♦ ❝æ♥❣ t❤ù❝ ð ✭✷✳✷✽✮✳ ❉♦ ✤â✱ tø ✭✷✳✷✻✮✱ t❛ s✉② r❛ Q\P = I(xk , bk )\P = I1 (xk , bk , x∗k ) ❑❤✐ ✤â✱ (ηk∗ )Q\P = (ηk∗ )I1 (xk ,bk ,x∗k ) ✈ỵ✐ ♠å✐ ∈ N ❇➡♥❣ ❇ê ✤➲ ✷✳✸✱ t❛ ❝â ❝→❝ t➟♣ ❤ñ♣ AQ,P ✈➔ BQ,P ✤â♥❣ ②➳✉∗ ✳ ❈❤♦ → ∞✱ w tø ❜❛♦ ❤➔♠ t❤ù❝ ✤➛✉ t✐➯♥ tr♦♥❣ ✭✷✳✷✼✮ ✈➔ (u∗k , vk ) −→ (x∗ , v) t❛ ❝â ∗ (x∗ , v) ∈ AQ,P × BQ,P ✭✷✳✷✾✮ w∗ ❱➻ (u∗k , ηk∗ , vk ) −→ (x∗ , b∗ , v), t❛ ❝â (ηk∗ )i → b∗i ✈ỵ✐ i ∈ T ❉♦ ✤â✱ ❝❤♦ → ∞ ✈➔ tø ✭✷✳✷✼✮✱ t❛ t➻♠ ✤÷đ❝ b∗i a∗i , b∗Q¯ = 0, b∗Q\P x∗ = − ✭✷✳✸✵✮ i∈Q ❇➡♥❣ ❝→❝❤ ❝❤å♥ P ✈➔ Q s❛♦ ❝❤♦ P ⊂ Q ⊂ I ✈➔ P ∈ I ❚ø ✭✷✳✷✾✮✱ ✭✷✳✸✵✮ ✈➔ ✭✷✳✷✷✮✱ t❛ ❝â (x∗ , b∗ , v) ∈ Σ(¯ x, ¯b, x¯∗ ) ❙✉② r❛ N (¯ x, ¯b, x¯∗ ); gph F ⊂ Σ(¯ x, ¯b, x¯∗ ) ✣➸ t➻♠ ✤÷đ❝ ❜❛♦ ❤➔♠ t❤ù❝ ✤➛✉ t✐➯♥ tr♦♥❣ ✭✷✳✷✹✮✱ ❝❤♦ (x∗ , b∗ , v) ∈ Σ0 (¯ x, ¯b, x¯∗ ) ❈❤å♥ P, Q ✈ỵ✐ P ⊂ Q ⊂ I, P ∈ I, ✈➔ FQ (¯b) = ∅ s❛♦ ❝❤♦ (x∗ , v) ∈ AQ,P × BQ,P , x∗ = − b∗i a∗i , b∗Q¯ = 0, b∗Q\J i∈Q ✸✹ ✭✷✳✸✶✮ ❈è ✤à♥❤ x ∈ FQ (¯b)✳ t ữợ xk = k x + (1 − k −1 )¯ x ❍✐➸♥ ♥❤✐➯♥ xk ∈ FQ (¯b) ✈➔ xk → x ¯ ❦❤✐ k → ∞ ❱ỵ✐ ♠å✐ k, ✤➦t bk = ¯b t❛ ❝â I(xk , bk ) = Q ❚ø P ∈ I, ❦➨♦ t❤❡♦ P = ∅ ✈➔ x¯∗ = λi a∗i ✈ỵ✐ λi 0, i ∈ P i∈P ❚❛ ✤➦t x∗k = (λi + k −1 )a∗i ∈ N (xk ; Θ(bk )) = F(xk , bk ) ✭✷✳✸✷✮ i∈P gph F x, ¯b, x¯∗ ) ❚ø ✭✷✳✸✷✮ ❦➨♦ ¯∗ ❦❤✐ k → ∞, t❛ ❝â (xk , bk , x∗k ) −−−→ (¯ ❚ø x∗k → x t❤❡♦ Q\P ⊂ I1 (xk , bk , x∗k ) ⊂ I(xk , bk ) = Q ✈ỵ✐ ♠å✐ k ∈ N ❇➡♥❣ ❝→❝❤ ①➨t ♠ët ❞➣② ❝õ❛ {k}, ♥➳✉ ❝➛♥ t❤✐➳t✱ t❛ ❝â t❤➸ ❣✐↔ sû r➡♥❣ I1 (xk , bk , x∗k ) = I1 ✈ỵ✐ ♠å✐ k ∈ N ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ ❜❛♦ ❤➔♠ t❤ù❝ I1 ⊂ Q\J ❚❤➟t ✈➟②✱ ❧➜② ❜➜t ❦➻ i ∈ I1 ❱ỵ✐ ♠å✐ k N, tỗ t àki ợ j Q\{i} t❤ä❛ ♠➣♥ x∗k = µkj a∗j ∈ ♣♦s a∗j : j Q\{i} , jQ ợ àki := ❚❤❡♦ ❇ê ✤➲ ✶✳✶✱ t❛ t➻♠ ✤÷đ❝ ♠ët t➟♣ ❤đ♣ K ⊂ Q\{i} ✈➔ ♠ët ❞➣② {k } ❝õ❛ {k} s❛♦ ❝❤♦ {a∗j : j ∈ K} ❧➔ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ ✈➔ x∗k ∈ ♣♦s{a∗j : j ∈ K} ✈ỵ✐ ♠å✐ ∈ N ❚ø {a∗j : j ∈ K} ❧➔ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ ✈➔ x∗k → x ¯∗ , t❛ t➻♠ ✤÷đ❝ x¯∗ ∈ ♣♦s{a∗j : j ∈ K} ✭✷✳✸✸✮ ❱ỵ✐ i ∈ K, t❤❡♦ ✤à♥❤ ♥❣❤➽❛ I1 (¯ x, ¯b, x¯∗ ) ✈➔ ✭✷✳✸✸✮✱ t❛ ❝â i ∈ I1 (¯ x, ¯b, x¯∗ ) ❱➻ J = I\I1 (¯ x, ¯b, x¯∗ ) ❦➨♦ t❤❡♦ i ∈ Q\J ❉♦ ✤â✱ ❜❛♦ ❤➔♠ I1 ⊂ Q\J ❧➔ ❧✉æ♥ ✤ó♥❣✳ ❚❛ ❝â I1 (xk , bk , x∗k ) = I1 ⊂ Q\J ✈ỵ✐ ♠é✐ k ∈ N, tø ✭✷✳✸✶✮ ❦➨♦ t❤❡♦ (x∗ , v) ∈ AQ,P × BQ,P , x∗ = − b∗i a∗i , b∗Q¯ = 0, b∗I˜1 i∈Q ✸✺ ❚❤❡♦ ❇ê ✤➲ ✷✳✺✱ t❛ s✉② r❛ (x∗ , b∗ , v) ∈ N (xk , bk , x∗k ); gph F ✈ỵ✐ ♠å✐ gph F k ∈ N ❑➳t ❤ñ♣ ✤✐➲✉ ♥➔② ✈➔ (xk , bk , x∗ ) −−−→ (¯ x, ¯b, x¯∗ ) ❦➨♦ t❤❡♦ k (x∗ , b∗ , v) ∈ N (¯ x, ¯b, x¯∗ ); gph F ❉♦ ✤â✱ t❛ ❝â Σ0 (¯ x, ¯b, x¯∗ ) ⊂ N (¯ x, ¯b, x¯∗ ); gph F ❚❛ ✤✐ ❝❤ù♥❣ ♠✐♥❤ ❝❤✐➲✉ ♥❣÷đ❝ ❧↕✐ ❝õ❛ ❜❛♦ ❤➔♠ tr♦♥❣ ✭✷✳✷✺✮✳ ●✐↔ t❤✐➳t r➡♥❣ x ¯∗ = ❈è ✤à♥❤ λ = (λi )i∈I ∈ Ξ(¯ x, ¯b, x¯∗ ) ợ ( x, b, x ) ữủ ổ t❤ù❝ ✭✷✳✶✮✳ ❈❤å♥ Q = I ✈➔ P = {i ∈ I : λi > 0}, t❛ ❝â I1 = I\J = Q\J ❚❤❡♦ ❇ê ✤➲ ✷✳✺✱ N (¯ x, ¯b, x¯∗ ); gph F     ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ bi , bQ¯ = 0, bI1 = (x , b , v) : (x , v) ∈ AQ,P × BQ,P , x = −   i∈Q     ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ bi , bQ¯ = 0, bQ\J = (x , b , v) : (x , v) ∈ AQ,P × BQ,P , x = −   i∈Q ✭✷✳✸✹✮ ❱➻ x ¯∗ = 0, t❛ s✉② r❛ P = ∅ ữ ỵ r P Q I, P I ◆❣♦➔✐ r❛✱ t❛ ❝â FQ (¯b) = ∅ ✈➻ x ¯ ∈ FQ (¯b) ❉♦ ✤â✱ N (¯ x, ¯b, x¯∗ ); gph F ⊂ Σ0 (¯ x, ¯b, x¯∗ ) ❚❛ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ✤à♥❤ ❧➼ tr➯♥✳ ✣➸ ữợ ❝❤♦ t❤➜② ❜❛♦ ❤➔♠ t❤ù❝ ✤➛✉ t✐➯♥ ❝õ❛ ✭✷✳✷✺✮ ♥â✐ ❝❤✉♥❣ ❧➔ ♥❣➦t✳ ✣✐➲✉ ♥➔② ❝â ♥❣❤➽❛ ❧➔ ✤→♥❤ ❣✐→ ữợ ( x, b, x ) N ( x, ¯b, x¯∗ ); gph F tr♦♥❣ ✣à♥❤ ❧➼ ✷✳✹ ❝❤♦ N (¯ x, ¯b, x¯∗ ); gph F ❧➔ tèt ❤ì♥ s♦ ✈ỵ✐ ✤→♥❤ ❣✐→ N (¯ x, ¯b, x¯∗ ); gph F ⊂ N (¯ x, ¯b, x¯∗ ); gph F ❱➼ ❞ư ✷✳✸✳ ❱ỵ✐ X, T ✈➔ {a∗i : i ∈ T } ❣✐è♥❣ ♥❤÷ tr♦♥❣ ❱➼ ❞ö ✷✳✶✳ ❱✐➺❝ t➼♥❤ t♦→♥ tr♦♥❣ ❱➼ ❞ö ✷✳✶ ✤↔♠ ❜↔♦ r➡♥❣ (¯ x, ¯b, x¯∗ ) ∈ gph F ✈ỵ✐ ¯b = (0, 0, 0) ∈ R3 , x¯ = (0, 0) ∈ X, ✸✻ ✈➔ x ¯∗ = (0, α) ✈ỵ✐ α > ❚❛ ❝â I = I(¯ x, ¯b) = {1, 2, 3}, I¯ = T \I = ∅, I1 = I1 (¯ x, ¯b, x¯∗ ) = {1, 3} ✈➔ J = I\I1 = {2} ❈â ♠ët ❝→❝❤ ❞✉② ♥❤➜t ❜✐➸✉ ❞✐➵♥ x ¯∗ = i∈I λi a∗i , λi 0✱ i ∈ I ❧➔ x¯∗ = (0, α) = 0(1, 0) + α(0, 1) + 0(1, 2) = 0a∗1 + αa∗2 + 0a∗3 ❱➻ ✈➟②✱ t❛ ❝â I = I(¯ x, ¯b, x¯∗ ) = {P ⊂ I : P = ∅, x¯∗ ∈ ♣♦s{a∗i : i ∈ P }} = {P ⊂ I : ∈ P } ❉♦ ✤â✱ P ✈➔ Q t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ P ⊂ Q ⊂ I ✈➔ P ∈ I ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ♠ët tr♦♥❣ ❝→❝ tr÷í♥❣ ❤đ♣ s❛✉ ✤↕t ✤÷đ❝✿ ✭❛✮ Q = {1, 2, 3} ✈➔ P = {1, 2, 3}, (0, 0) ∈ FQ (¯b) = ∅; ✭❜✮ Q = {1, 2, 3} ✈➔ P = {1, 2}, (0, 0) ∈ FQ (¯b) = ∅; ✭❝✮ Q = {1, 2, 3} ✈➔ P = {2, 3}, (0, 0) ∈ FQ (¯b) = ∅; ✭❞✮ Q = {1, 2, 3} ✈➔ P = {2}, (0, 0) ∈ FQ (¯b) = ∅; ✭❡✮ Q = {1, 2} ✈➔ P = {1, 2}, FQ (¯b) = ∅; ✭❢✮ Q = {1, 2} ✈➔ P = {2}, FQ (¯b) = ∅; ✭❣✮ Q = {2, 3} ✈➔ P = {2, 3}, FQ (¯b) = ∅; ✭❤✮ Q = {2, 3} ✈➔ P = {2}, FQ (¯b) = ∅; ✭✐✮ Q = {2} ✈➔ P = {2}, (−1, 0) ∈ FQ (¯b) = ∅ ❱ỵ✐ ♠é✐ t ∈ Γ := {a, b, c, d, e, f, g, h, i}, ✤à♥❤ ♥❣❤➽❛ P, Q ♥❤÷ tr♦♥❣ tr÷í♥❣ ❤đ♣ (t) ð tr➯♥ ✈➔ ✤➦t ✸✼ Λ(t) Λ(t)   = (x∗ , b∗ , v) : (x∗ , v) ∈ AQ,P × BQ,P , x∗ = − b∗i a∗i , b∗Q¯ = 0, b∗Q\P  i∈Q   = (x∗ , b∗ , v) : (x∗ , v) ∈ AQ,P × BQ,P , x∗ = − b∗i a∗i , b∗Q¯ = 0, b∗Q\J  i∈Q Λ0(t) = Λ(t) ∅ ♥➳✉ FQ (¯b) = ∅, ♥➳✉ FQ (¯b) = ∅ ❚ø x ¯∗ = (0, α) = 0R2 , t❛ ❝â ∅ ∈ I(¯ x, ¯b, x¯∗ ) ❚❛ t❤➜② r➡♥❣ {a∗i : i ∈ P } ❧➔ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ P ✤÷đ❝ tr trữớ ủ (t) ợ t \{a} ✤â✱ Λ(t) ✈➔ Σ0 (¯ x, ¯b, x¯∗ ) = Σ(¯ x, ¯b, x¯∗ ) = Λ0(t) ✭✷✳✸✺✮ t∈Γ t∈Γ\{a} ❈→❝ t➟♣ ❤đ♣ Λ(t) ✈➔ Λ0(t) ✤÷đ❝ t➼♥❤ t♦→♥ t❤❡♦ t ∈ Γ ♥❤÷ s❛✉✿ ✶✳ ◆➳✉ t = a, t❤➻ Q = {1, 2, 3} ✈➔ P = {1, 2, 3}✳ ❚❛ ❝â AQ,P = s♣❛♥{a∗i : i ∈ P } = R2 , BQ,P = (AQ,P )∗ = {0R2 } ú ỵ r FQ (b) = ❈❤♦ x∗ ∈ AQ,P s❛♦ ❝❤♦ x∗ = − b∗Q\J ❚❤➻✱ x∗ = −b∗1 a∗1 − b∗2 a∗2 − b∗3 a∗3 = (−b∗1 − b∗3 , −b∗2 − 2b∗3 ) ✈➔ b∗1 ❉♦ ✤â✱ b∗ ∈ R− × R × R− ✈➔ x∗ ∈ R+ × R ❱➻ ✈➟②✱ Λ0(a) = (R+ × R) × (R− × R × R− ) × {02R } ✸✽ ∗ ∗ i∈Q bi 0, b∗3 ✈➔ 0   ,    ,  ✷✳ ◆➳✉ t = b, t❤➻ Q = {1, 2, 3} ✈➔ P = {1, 2} ❚❛ ❝â AQ,P = s♣❛♥{a∗1 , a∗2 } + ♣♦s{a∗3 } = R2 , ❈❤♦ x∗ ∈ AQ,P s❛♦ ❝❤♦ x∗ = − ∗ ∗ i∈Q bi BQ,P = (AQ,P )∗ = {0R2 } ✈➔ b∗Q\P ❚❤➻✱ x∗ = −b∗1 a∗1 − b∗2 a∗2 − b∗3 a∗3 = (−b∗1 − b∗3 , −b∗2 − 2b∗3 ) ✈➔ b∗3 ❉♦ ✤â✱ b∗ ∈ R2 × R− ✈➔ x∗ ∈ R2 ❱➻ ✈➟②✱ Λ(b) = R2 × (R2 ì R ) ì {0R2 } ú ỵ r FQ (¯b) = ∅ ❈❤♦ x∗ ∈ AQ,P s❛♦ ❝❤♦ x∗ = − b∗Q\J ❚❤➻✱ x∗ = (−b∗1 − b∗3 , −b∗2 − 2b∗3 ) ✈➔ b∗1 0, b∗3 ∗ ∗ i∈Q bi ✈➔ ❉♦ ✤â✱ b∗ ∈ R− × R × R− ✈➔ x∗ ∈ R+ × R ❱➻ ✈➟②✱ Λ0(b) = (R+ × R) × (R− × R × R− ) × {0R2 } ✣è✐ ợ trữớ ủ t = c, , t = h t❛ ❧➔♠ t÷ì♥❣ tü ✤➸ t❤✉ ✤÷đ❝ ❝→❝ ❦➳t q✉↔ s❛✉✿ ✸✳ ❱ỵ✐ t = c✱ Λ(c) = R2 × (R− × R2 ) × {0R2 }, Λ0(c) = (R+ × R) × (R− × R × R ) ì {0R2 } ợ t = d, Λ(d) = Λ0(d) = (R+ × R) × (R− × R × R− ) × R− × {0} ✺✳ ợ t = e, (e) = R2 ì R2 ì {0} × {0R2 }, ✸✾ Λ0(e) = ∅ ✻✳ ợ t = f, (f ) = (R+ ì R) × R− × R × {0} × R− × {0} , Λ0(f ) = ∅ ✼✳ ❱ỵ✐ t = ❣, Λ(❣) = R2 × {0} × R2 × {0R2 }, Λ0(❣) = ∅ ✽✳ ❱ỵ✐ t = h, Λ(h) = (R+ × R) × {0} × R × R− × R ì {0} , 0(h) = r trữớ ủ t = i ❧➔ tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t✳ ❈→❝❤ ❣✐↔✐ tt ữợ t = i, t❤➻ Q = {2} ✈➔ P = {2} ❚❛ ❝â AQ,P = s♣❛♥{a∗2 } = {0} × R, ❈❤♦ x∗ ∈ AQ,P s❛♦ ❝❤♦ x∗ = − BQ,P = (AQ,P )∗ = R × {0} ∗ ∗ i∈Q bi , x∗ = −b∗2 a∗2 = (0, −b∗2 ), b∗Q¯ = 0, ✈➔ b∗Q\P ❚❤➻✱ b∗1 = b∗3 = ❉♦ ✤â✱ b∗ ∈ {0} × R × {0} ✈➔ x∗ ∈ {0} × R ❱➻ ✈➟②✱ Λ(i) = {0} × R × {0} × R × {0} × R × {0} ❚❛ ❝â FQ (¯b) = ∅ ✈➔ Q\P = Q\J, ❞♦ ✤â Λ0(i) = Λ(i) = {0} × R × {0} × R × {0} × R × {0} ❚ø ❝→❝ ❦➳t q✉↔ tr➯♥ ✈➔ ✭✷✳✸✺✮✱ t❛ t❤✉ ✤÷đ❝ ❝ỉ♥❣ t❤ù❝ t➼♥❤ ❝❤➼♥❤ ①→❝ ❝❤♦ Σ(¯ x, ¯b, x¯∗ ) ✈➔ Σ0 (¯ x, ¯b, x¯∗ ) ❚ø x¯∗ = 0, t❤❡♦ ✭✷✳✷✺✮✱ t❛ ❝â N (¯ x, ¯b, x¯∗ ); gph F ⊂ Σ0 (¯ x, ¯b, x¯∗ ) ❚ø ✭✷✳✸✺✮✱ Λ0(i) = {0} × R × {0} × R × {0} × R × {0} ⊂ Σ0 (¯ x, ¯b, x¯∗ ) ✹✵ ❉ü❛ ✈➔♦ ❝❤ù♥❣ ♠✐♥❤ tr♦♥❣ ❱➼ ❞ö ✷✳✸✱ N (¯ x, ¯b, x¯∗ ); gph F = (R+ × R) × (R− × R × R− ) × R− × {0} x, ¯b, x¯∗ ); gph F , t❛ s✉② r❛ ❚ø Λ0(i) ⊂ N (¯ N (¯ x, ¯b, x¯∗ ); gph F Σ0 (¯ x, ¯b, x¯∗ ) ❚ø ✣à♥❤ ❧➼ ✷✳✹✱ t❛ ❞➵ ❞➔♥❣ t➻♠ ✤÷đ❝ tr ữợ ố r D∗ F(x, b, x∗ ) : X ∗∗ ⇒ X ∗ × Rm ❝õ❛ F t↕✐ ♠ët ✤✐➸♠ (x, b, x∗ ) ∈ gph F ✣➦t Ω(x, b, x∗ )(v) = (u∗ , η ∗ ) ∈ X ∗ × Rm : (u∗ , η ∗ , −v) ∈ Σ(x, b, x∗ ) ✈➔ Ω0 (x, b, x∗ )(v) = (u∗ , η ∗ ) ∈ X ∗ × Rm : (u∗ , η ∗ , −v) ∈ Σ0 (x, b, x∗ ) , t❛ ❝â Ω0 (x, b, x∗ )(v) ⊂ D∗ F(x, b, x∗ )(v) ⊂ Ω(x, b, x∗ )(v) ✈ỵ✐ ♠å✐ v ∈ X ∗∗ ✹✶ ❑➳t ❧✉➟♥ ❝õ❛ ❈❤÷ì♥❣ ✷ ❑➳t q✉↔ ❝❤➼♥❤ ❝õ❛ ❝❤÷ì♥❣ ỗ r ổ tự t ✤è✐ ✤↕♦ ❤➔♠ ❋r➨❝❤❡t ❝õ❛ →♥❤ ①↕ ♥â♥ ♣❤→♣ t✉②➳♥ t ỗ tổ q r tr ữợ ❝❤♦ ✤è✐ ✤↕♦ ❤➔♠ ▼♦r✲ ❞✉❦❤♦✈✐❝❤ ❝õ❛ →♥❤ ①↕ ♥â♥ t t ỗ tổ q ❧➼ ✷✳✹✳ ✲ ❈❤÷ì♥❣ ♥➔② ❝á♥ tr➻♥❤ ❜➔② ❝→❝ ❱➼ ❞ö ✷✳ ✷ ✈➔ ❱➼ ❞ö ✷✳ ✸ ♠✐♥❤ ❤å❛ ❝❤♦ ✈✐➺❝ t➼♥❤ t♦→♥ ❝❤➼♥❤ ①→❝ ✤è✐ ✤↕♦ ❤➔♠ ❋r➨❝❤❡t tr ữợ ố ❤➔♠ ▼♦r❞✉❦❤♦✈✐❝❤ ❝õ❛ →♥❤ ①↕ ♥â♥ ♣❤→♣ t✉②➳♥ ❝õ❛ t➟♣ ỗ ❝õ❛ ❧✉➟♥ ✈➠♥ ❧➔ tr➻♥❤ ❜➔② ♠ët ❝→❝❤ ❝â ❤➺ t❤è♥❣ ❝→❝ ❦➳t q✉↔ tr♦♥❣ ❜➔✐ ❜→♦ ❬✽❪ ✈➲ →♥❤ õ t t ỗ õ ♥❤✐➵✉ t✉②➳♥ t➼♥❤✳ ❈ö t❤➸✱ ✲ ❚r➻♥❤ ❜➔② ❝→❝ ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ✈➲ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ✤➦❝ ❜✐➺t✱ t➟♣ ỗ õ t t ỗ ❞✐➺♥✱ ♥â♥ ♣❤→♣ t✉②➳♥ ❋r➨❝❤❡t✱ ♥â♥ ♣❤→♣ t✉②➳♥ ▼♦r❞✉❦❤♦✈✐❝❤ ❝õ❛ ♠ët t➟♣ ❤ñ♣ ✈➔ ❝→❝ ✤è✐ ✤↕♦ ❤➔♠ ✭✤è✐ ✤↕♦ ❤➔♠ ❋r➨❝❤❡t✱ ✤è✐ ✤↕♦ ❤➔♠ ▼♦r❞✉❦❤♦✈✐❝❤✮ ❝õ❛ →♥❤ ①↕ ✤❛ trà✳ ✲ ❚r➻♥❤ ❜➔② ❝æ♥❣ t❤ù❝ t➼♥❤ t♦→♥ ✤è✐ ✤↕♦ ❤➔♠ ❋r➨❝❤❡t ❝õ❛ →♥❤ ①↕ ♥â♥ ♣❤→♣ t✉②➳♥ ❝õ❛ t➟♣ ỗ r tr ữợ ố r õ t t ỗ ❈â t❤➸ ♣❤→t tr✐➸♥ ❝→❝ ❦➳t q✉↔ ❝õ❛ ❧✉➟♥ ✈➠♥ ♥❤÷ s❛✉✿ ✲ ❚➻♠ ❝→❝ ❝ỉ♥❣ t❤ù❝ t➼♥❤ ✤è✐ ✤↕♦ ❤➔♠ ❋r➨❝❤❡t✱ ✤è✐ ✤↕♦ ❤➔♠ ▼♦r✲ ❞✉❦❤♦✈✐❝❤ ❝õ❛ →♥❤ ①↕ õ t t ỗ õ ♣❤✐ t✉②➳♥✳ ✲ ❚➻♠ ❝→❝ ❝æ♥❣ t❤ù❝ t➼♥❤ ✤è✐ ✤↕♦ ❤➔♠ ❋r➨❝❤❡t✱ ✤è✐ ✤↕♦ ❤➔♠ ▼♦r✲ ❞✉❦❤♦✈✐❝❤ ❝õ❛ →♥❤ ①↕ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✤✐➲✉ ❦❤✐➸♥ tè✐ ÷✉ ❝❤ù❛ t❤❛♠ sè ✤÷đ❝ ❝❤♦ ❜ð✐ ♣❤÷ì♥❣ tr➻♥❤ tr↕♥❣ t❤→✐ ♣❤✐ t✉②➳♥✳ ✹✸ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬❆❪ ❚➔✐ ❧✐➺✉ t✐➳♥❣ ❱✐➺t ❬✶❪ ◆❣✉②➵♥ ✣æ♥❣ ❨➯♥ ✭✷✵✵✼✮✱ ●✐→♦ tr➻♥❤ ●✐↔✐ t➼❝❤ ✤❛ trà✱ ◆❳❇ ❑❤♦❛ ❤å❝ tü ♥❤✐➯♥ ✈➔ ❈æ♥❣ ♥❣❤➺✱ ❍➔ ◆ë✐✳ ❬❇❪ ❚➔✐ ❧✐➺✉ t✐➳♥❣ ❆♥❤ ❬✷❪ ❉✳ ❇❛rt❧ ✭✷✵✵✽✮✱ ❆ s❤♦rt ❛❧❣❡❜r❛✐❝ ♣r♦♦❢ ♦❢ t❤❡ ❋❛r❦❛s ❧❡♠♠❛✱ ❙■❆▼ ❏✳ ❖♣t✐♠✳ ✱ ✷✸✹✲✷✸✾✳ ✶✾ ❬✸❪ ❘✳ ❍❡♥r✐♦♥✱ ❇✳❙ ▼♦r❞✉❦❤♦✈✐❝❤✱ ◆✳▼✳ ◆❛♠ ✭✷✵✶✵✮✱ ❙❡❝♦♥❞✲♦r❞❡r ❛♥❛❧②s✐s ♦❢ ♣♦❧②❤❡❞r❛❧ s②st❡♠s ✐♥ ❢✐♥✐t❡ ❛♥❞ ✐♥❢✐♥✐t❡ ❞✐♠❡♥s✐♦♥s ✇✐t❤ ❛♣♣❧✐❝❛t✐♦♥ t♦ r♦❜✉st st❛❜✐❧✐t② ♦❢ ✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t✐❡s✱ ❙■❆▼ ❏✳ ❖♣t✐♠✳ ✱ ✷✶✾✾✲✷✷✷✼✳ ✷✵ ❬✹❪ ❇✳❙✳ ▼♦r❞✉❦❤♦✈✐❝❤✱ ❱❛r✐❛t✐♦♥❛❧ ❆♥❛❧②s✐s ❛♥❞ ●❡♥❡r❛❧✐③❡❞ ❉✐❢❢❡r❡♥✲ t✐❛t✐♦♥✱ ❱♦❧✳ ■✿ ❇❛s✐❝ ❚❤❡♦r②✱ ❱♦❧✳ ■■✿ ❆♣♣❧✐❝❛t✐♦♥s✱ ❙♣r✐♥❣❡r✱ ❇❡r❧✐♥✳ ❬✺❪ ◆✳▼✳ ◆❛♠ ✭✷✵✵✻✮✱ ❈♦❞❡r✐✈❛t✐✈❡s ♦❢ ♥♦r♠❛❧ ♠❛♣♣✐♥❣s ❛♥❞ t❤❡ ▲✐♣✲ s❝❤✐t③✐❛♥ st❛❜✐❧✐t② ♦❢ ♣❛r❛♠❡tr✐❝ ✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t✐❡s✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧✳ ✱ ✶✻✼✻✲✶✻✽✾✳ ✼✸ ❬✻❪ ❍✳❚✳ P❤✉♥❣ ✭✷✵✵✵✮✱ ❖♥ t❤❡ ❧♦❝❛❧❧② ✉♥✐❢♦r♠ ♦♣❡♥♥❡ss ♦❢ ♣♦❧✐❤❡❞r❛❧ s❡ts✱ ❆❝t❛ ▼❛t❤✳ ❱✐❡t♥❛♠✳ ✱ ✷✼✸✲✷✽✹✳ ✷✺ ❬✼❪ ◆✳❚✳ ◗✉✐ ✭✷✵✶✶✮✱ ▲✐♥❡❛r❧② ♣❡rt✉r❜❡❞ ♣♦❧②❤❡❞r❛❧ ♥♦r♠❛❧ ❝♦♥❡ ♠❛♣✲ ♣✐♥❣s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧✳ ✱ ✶✻✼✻✲✶✻✽✾✳ ✼✹ ❬✽❪ ◆✳ ❚✳ ◗✉② ✭✷✵✶✶✮✱ ◆❡✇ r❡s✉❧ts ♦♥ ❧✐♥❡❛r❧② ♣❡rt✉r❜❡❞ ♣♦❧②❤❡❞r❛❧ ♥♦r✲ ♠❛❧ ❝♦♥❡ ♠❛♣♣✐♥❣s✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳ ✱ ✸✺✷✕✸✻✹✳ ✸✽✶ ✹✹ ❬✾❪ ❘✳ ❍❡♥r✐♦♥✱ ❇✳ ▼♦r❞✉❦❤♦✈✐❝❤✱ ◆✳▼✳ ◆❛♠ ✭✷✵✶✵✮✱ P♦❧②❤❡❞r❛❧ s②st❡♠s ✐♥ ❢✐♥✐t❡ ❛♥❞ ✐♥❢✐♥✐t❡ ❞✐♠❡♥s✐♠s ✇✐t❤ ❛♣♣❧✐❝❛t✐♦♥s t♦ r♦❜✉st st❛❜✐❧✐t② ♦❢ ✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t✐❡s✱ ❙■❆▼ ❏✳❖♣t✐♠✳ ✱ ✷✶✾✾✲✷✷✷✼✳ ✷✵ ✹✺ ... t tốt t tt t ỗ ỹ q t tố ữ ỡ tr ỵ t❤✉②➳t tè✐ ÷✉ ❧➔ ❜➔✐ t♦→♥ t➻♠ ❝ü❝ t✐➸✉ ❝õ❛ ởt số ữợ ởt số r t tố ữ õ ố q t tt ợ ởt sè ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥ ✤➳♥ tè✐ ÷✉✿ tø ❜➜t tự ỏ ữủ t ợ t t❤ỉ♥❣... ∗ : i ∈ T } ❧➔ ♠ët ❤➺ ✈❡❝t♦ ✈ỵ✐ T = {1, 2, , m} ❧➔ ♠ët t➟♣ ❤ú✉ ❤↕♥ ❝→❝ ❝❤➾ sè✱ t❛ ①➨t t ỗ ự t số (b) = x ∈ X : a∗i , x bi ✈ỵ✐ ♠å✐ i ∈ T ✭✶✳✼✮ ✈ỵ✐ b = (b1 , , bm ) ∈ Rm ❧➔ ❝→❝ t❤❛♠ sè✳ Ð ✤➙②✱... ♥❤✐➵✉ ✈➳ ♣❤↔✐ ❝õ❛ ❤➺ ❜➜t ✤➥♥❣ t❤ù❝ t✉②➳♥ t➼♥❤ a∗i , x bi , i ∈ T ✭✶✳✽✮ ❱ỵ✐ x ∈ Θ(b)✱ t➟♣ ❝→❝ ❝❤➾ số t tữỡ ự ợ (x, b) X × Rm ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ I(x, b) = {i ∈ T : a∗i , x = bi } ✭✶✳✾✮ ❈❤♦ I ⊂