Về một mở rộng của bao lồi trong mặt phẳng

❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ❱■◆❍ ◆●❯❨➍◆ ❍Ú❯ ❍■➌❯ ❱➋ ▼❐❚ ▼Ð ❘❐◆● ❈Õ❆ ❇❆❖ ▲➬■ ❚❘❖◆● ▼➄❚ P❍➃◆● ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ◆❣❤➺ ❆♥ ✲ ✷✵✶✾ ❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ❱■◆❍ ◆●❯❨➍◆ ❍Ú❯ ❍■➌❯ ❱➋ ▼❐❚ ▼Ð ❘❐◆● ❈Õ❆ ❇❆❖ ▲➬■ ❚❘❖◆● ▼➄❚ P❍➃◆● ❍➐◆❍ ❍➴❈ ❱⑨ ❚➷P➷ ▼➣ sè✿ ✽✳✹✻✳✵✶✳✵✺ ❈❤✉②➯♥ ♥❣➔♥❤ ✿ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ữợ ❚❤❛♥❤ ●✐❛♥❣ ◆❣❤➺ ❆♥ ✲ ✷✵✶✾ ▼ư❝ ❧ư❝ ▲í✐ ❝↔♠ ì♥ ✷ ▼Ð ✣❺❯ ✸ ◆❐■ ❉❯◆● ✺ ✶ ▼ët số tự ỗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ỡ ỗ tr t α✲✤❛ ❣✐→❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷ ❱➲ ❜➔✐ t♦→♥ t➼♥❤ ❜❛♦ α✲❧ã♠ ❝õ❛ ♠ët t➟♣ ✤✐➸♠ tr♦♥❣ ♠➦t ♣❤➥♥❣ ✶✻ α✲❧ã♠ ✷✳✶ ✣à♥❤ ♥❣❤➽❛ ❜❛♦ ✷✳✷ ❈→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❜❛♦ ✷✳✸ ❱➲ ❜➔✐ t♦→♥ t➼♥❤ ❜❛♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ α✲ ✶✼ ❧ã♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ α✲❧ã♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✸✳✶ ❱➲ ❜➔✐ t♦→♥ t➼♥❤ α− ✤❛ ❣✐→❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✸✳✷ ❱➲ ❜➔✐ t♦→♥ t➼♥❤ ❜❛♦ α−❧ã♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ❑➌❚ ▲❯❾◆ ✸✶ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ✸✶ ✶ ▲í✐ ❝↔♠ ì♥ ▲✉➟♥ ✈➠♥ ✤÷đ❝ t❤ü❝ ❤✐➺♥ t↕✐ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❱✐♥❤ ữợ sỹ ữợ ●✐❛♥❣✳ ❚→❝ ❣✐↔ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s ổ ữớ trỹ t ữợ ❞➝♥ ✈➔ ❣✐ó♣ ✤ï t→❝ ❣✐↔ r➜t ♥❤✐➲✉ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉✳ ◆❤➙♥ ❞à♣ ❜↔♦ ✈➺ ✤➲ t➔✐✱ t→❝ ❣✐↔ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ P●❙✳ ❚❙✳ P❤❛♥ ❚❤➔♥❤ ❆♥✱ ❱✐➺♥ ❚♦→♥ ❤å❝ ✕ ❱✐➺♥ ❍➔♥ ❧➙♠ ❑❍ ✈➔ ❈◆ ❱✐➺t ◆❛♠ ✤➣ ❝➜♣ ❤å❝ ❜ê♥❣ ❝❤♦ t→❝ ❣✐↔ ❤å❝ t➟♣✳ ❚→❝ ❣✐↔ ❝↔♠ ì♥ ♣❤á♥❣ ❙❛✉ ✤↕✐ ❤å❝✱ ❱✐➺♥ ❙÷ P❤↕♠ tü ♥❤✐➯♥ ✈➔ ❝↔♠ ì♥ ❝→❝ t❤➛②✱ ❝æ ❣✐→♦ tr♦♥❣ tê ❍➻♥❤ ❤å❝ ✤➣ ♥❤✐➺t t➻♥❤ ❣✐↔♥❣ ❞↕② ✈➔ ❣✐ó♣ ✤ï t→❝ ❣✐↔ tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ ❤å❝ t➟♣ ✈➔ ❤♦➔♥ t❤➔♥❤ ✤➲ t➔✐✳ ❈✉è✐ ❝ị♥❣ t→❝ ❣✐↔ ❝→♠ ì♥ ❣✐❛ ✤➻♥❤✱ t❤➛② ❝ỉ ❣✐→♦✱ ❜↕♥ ❜➧✱ ❝→❝ ❜➟❝ ♣❤ö ❤✉②♥❤ ❤å❝ s✐♥❤✱ ✤➦❝ ❜✐➺t ❧➔ ❝→❝ ❜↕♥ tr♦♥❣ ❧ỵ♣ ❈❛♦ ❤å❝ ❑✷✺ ✲ ❍➻♥❤ ❤å❝ ✈➔ tỉ♣ỉ ✤➣ ❝ë♥❣ t→❝✱ ❣✐ó♣ ✤ï ✈➔ ✤ë♥❣ ✈✐➯♥ t→❝ ❣✐↔ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉✳ ▼➦❝ ❞ò ✤➣ ❝â ♥❤✐➲✉ ❝è ❣➢♥❣✱ ♥❤÷♥❣ ❧✉➟♥ ✈➠♥ ❦❤ỉ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ ❤↕♥ ❝❤➳✱ t❤✐➳✉ sõt ú tổ rt ữủ ỳ ỵ ✤â♥❣ ❣â♣ ❝õ❛ ❝→❝ t❤➛②✱ ❝æ ❣✐→♦ ✈➔ ❝→❝ ❜↕♥ ❜➧ ✤➸ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ✷ ▼ð ỵ t ỗ ỗ q tở tr ỗ t t tr õ t t ỗ ởt tr ỳ t ỡ t t ỗ ♠ët t➟♣ ✤✐➸♠ ♥❤ä ♥❤➜t ❝❤ù❛ S✳ S tr♦♥❣ ♠➦t ỗ tt t ữủ ợ t t ỗ ởt t ỗ õ ù♥❣ ❞ư♥❣ tr♦♥❣ ❝→❝ ❧➽♥❤ ✈ü❝ ✤❛ ❞↕♥❣ ♥❤÷ ♥❤➟♥ ỷ ỵ tố tổ t ỵ ữớ ❉♦ ✤â✱ ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝ q✉❛♥ t➙♠ ♥❣❤✐➯♥ ❝ù✉ t rở ỗ t ỗ tr t tr ổ ữợ ự t ữủt ❝ù✉ ❝ơ♥❣ ♥❤÷ ✤➸ ❧➔♠ ♣❤♦♥❣ ♣❤ó t❤➯♠ ❝→❝ t➔✐ t rở ỗ tr ỡ sð ❜➔✐ ❜→♦ ✏ α✲❈♦♥❝❛✈❡ ❤✉❧❧✱ ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ❝♦♥✈❡① ❤✉❧❧✑ ❝õ❛ ❝→❝ t→❝ ❣✐↔ ❙✳ ❆s❛❡❡❞✐✱ ❋✳ ❉✐❞❡❤✈❛r ✈➔ ❆✳ ▼♦❤❛❞❡s ✤➠♥❣ tr➯♥ t↕♣ ❝❤➼ ❚❤❡♦r❡t✐❝❛❧ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡ ♥➠♠ ✷✵✶✼✱ ❝❤ó♥❣ tỉ✐ ❧ü❛ ❝❤å♥ ✤➲ t➔✐✿ ✏❱➲ ♠ët ♠ð rở ỗ tr t ❤➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ❧✐➯♥ q✉❛♥ ✤➳♥ ✤➲ t➔✐ ▲➔ ♠ët ỡ ỗ ữủ tr tr t ữ ỗ ❝õ❛ ♠ët t➟♣ ✤✐➸♠ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❒❝❧✐t ❧➔ t➟♣ ỗ t ự t õ õ r tr t ỗ ởt t ỗ ọ t ự S S ó t ữủ ợ t t ữợ t t ổ ỗ tr♦♥❣ ❬✺❪ ✈➔ ✤÷đ❝ ♠ð rë♥❣ tr♦♥❣ ❬✾❪✳ ❇❛♦ ❧ã♠ ✸ ✹ ❧➔ ♠ët ❝ỉ♥❣ ❝ư ✤÷đ❝ sû ❞ư♥❣ ✤➸ t➼♥❤ t♦→♥ ❤➻♥❤ ❜❛♦ ❝õ❛ ♠ët t➟♣ ❤ñ♣ ✤✐➸♠ ❜➡♥❣ t r ỗ ổ ỗ ❞✐➺♥ ❝❤♦ ♠✐➲♥ ❜à ❝❤✐➳♠ ❜ð✐ ❝→❝ ✤✐➸♠ ✤➣ ❝❤♦✳ ợ ỗ ú t ổ õ ❞✉② ♥❤➜t ❝❤♦ ❜❛♦ ❧ã♠ ❝õ❛ ♠ët t➟♣ ✤✐➸♠ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❒❝❧✐t ♥â✐ ❝❤✉♥❣✳ ❚✉② ♥❤✐➯♥✱ tr♦♥❣ ♠➦t ♣❤➥♥❣✱ ❜❛♦ ❧ã♠ ❝õ❛ ♠ët t➟♣ ✤✐➸♠ ❝â t❤➸ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❧➔ ✤❛ ❣✐→❝ ❝â ❞✐➺♥ t➼❝❤ ♥❤ä ♥❤➜t ❝❤ù❛ t➟♣ ✤✐➸♠ ✤â✱ ✈➔ ❝❤♦ ♣❤➨♣ ❝→❝ ❣â❝ tr♦♥❣ ❝õ❛ ✤❛ ❣✐→❝ ❧➔ ❜➜t ❦ý✳ ❚r♦♥❣ ❬✶✶❪✱ ❏✐♥✲❙❡♦ P✳ ❛♥❞ ❙❡✲❏♦♥❣ ❖✳ ✤➣ ❝❤➾ r❛ r➡♥❣ ✤➸ ♥❤➟♥ ❞✐➺♥ ❤➻♥❤ ❜❛♦ ♠ët t➟♣ ✤✐➸♠✱ ♠✐➲♥ ❜❛♦ ❧ã♠ ❧➔ ❝❤➼♥❤ ①→❝ ỳ ỡ ỗ ỗ ổ →♥❤ ✤➛② ✤õ ❝→❝ ✤➦❝ tr÷♥❣ ❤➻♥❤ ❤å❝ ❝õ❛ t➟♣ ✤✐➸♠✱ ✈➔ ❜❛♦ ❧ã♠ ❧➔ ♠ët ❧ü❛ ❝❤å♥ tèt ❤ì♥ ✤➸ ✤→♥❤ ❣✐→ t➼♥❤ ❤➻♥❤ ❤å❝✳ ❚➼♥❤ t♦→♥ ❜❛♦ ❧ã♠ ❧➔ ♠ët ♣❤÷ì♥❣ ♣❤→♣ tèt ❤ì♥ ✤➸ ♥➢♠ ❜➢t ❤➻♥❤ ❞↕♥❣ ❝❤➼♥❤ ①→❝ ❝õ❛ ❜➲ ♠➦t t➟♣ ✤✐➸♠ ✭t➟♣ ❞ú ❧✐➺✉✮❀ t✉② ♥❤✐➯♥✱ ✈✐➺❝ t➼♥❤ t♦→♥ ❜❛♦ ❧ã♠ ❧➔ ❦❤â ❦❤➠♥ ❤ì♥✳ ❚r♦♥❣ ❬✶✶❪✱ ♠ët t❤✉➟t t♦→♥ ✤÷đ❝ tr➻♥❤ ❜➔② ✤➸ t➼♥❤ ❜❛♦ ❧ã♠ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ d ❝❤✐➲✉✳ ❇❛♦ ❧ã♠ ❝â t➼♥❤ ù♥❣ ❞ö♥❣ ❤✐➺✉ q✉↔ tr♦♥❣ ❝→❝ ♥❣➔♥❤ ♥❤÷ t→✐ t↕♦ ❤➻♥❤ ❞↕♥❣✱ t➼♥❤ t♦→♥ ✈➟t ❧✐➺✉ ✭①❡♠ tố tổ t ỵ ❧♦↕✐ sè ❧✐➺✉ ✭①❡♠ ❬✹❪✮✱✳✳✳ ❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔②✱ ❝❤ó♥❣ tổ t t ỗ ó ❝õ❛ ♠ët t➟♣ ✤✐➸♠ tr♦♥❣ ♠➦t ♣❤➥♥❣✳ ❈❤ó♥❣ tỉ✐ tr➻♥❤ ởt rở ỗ ❜❛♦ α✲❧ã♠✳ ❇❛♦ α✲❧ã♠ ❝õ❛ ♠ët t➟♣ ✤✐➸♠ tr♦♥❣ ♠➦t ởt ổ ỗ ợ r õ ữợ t ọ t ợ ó = tữỡ ữỡ ợ ỗ ❝ù✉ ▲✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② ✈➲ ❜❛♦ α✲❧ã♠✱ ♠ët ♠ð rở ỗ r t t ❦❤→✐ ♥✐➺♠ ♥➔② ✈➔ ❝→❝ ❦❤→✐ ♥✐➺♠ ❧✐➯♥ q✉❛♥✱ tr➻♥❤ ❜➔② t❤✉➟t t♦→♥ ❝❤✉②➸♥ ✤ê✐ tø ❜➔✐ t♦→♥ ✤➣ ❜✐➳t ✈➲ ❜➔✐ t♦→♥ t➼♥❤ ❜❛♦ ❝õ❛ t❤✉➟t t♦→♥ t➼♥❤ ❜❛♦ ó ó tứ õ sỹ tỗ t tr t❤í✐ ❣✐❛♥ ❤ú✉ ❤↕♥✳ ✹✳ ◆❤✐➺♠ ✈ư ♥❣❤✐➯♥ ❝ù✉ ✲ ❍➺ t❤è♥❣ ❝→❝ ❦❤→✐ ♥✐➺♠✱ ❝→❝ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ t ỗ ỗ ỡ ó ự tt t t ởt số ỵ ❧✐➯♥ q✉❛♥ ✤➳♥ ❝→❝ ❦❤→✐ ♥✐➺♠ tr➯♥✳ ✺ ✲ ❚r➻♥❤ ❜➔② ❝❤✐ t✐➳t t❤✉➟t t♦→♥ ❝❤✉②➸♥ ✤ê✐ tø ❜➔✐ t♦→♥ tê♥❣ t➟♣ ❝♦♥ ✤➣ ❜✐➳t ✈➲ ❜➔✐ t♦→♥ t➼♥❤ α✲✤❛ ởt t ợ t trữợ ✈➔ t❤✉➟t t♦→♥ ❝❤✉②➸♥ ✤ê✐ tø ❜➔✐ t♦→♥ ♣❤➙♥ ♥❤â♠ t❛♠ ❣✐→❝ ✈➲ ❜➔✐ t♦→♥ t➼♥❤ ❜❛♦ t➟♣ ✤✐➸♠ tr♦♥❣ ♠➦t ♣❤➥♥❣ ✭✈ỵ✐ α✲❧ã♠ ❝õ❛ ♠ët < α < π ✮✳ ✺✳ ✣è✐ t÷đ♥❣ ✈➔ ♣❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉ ố tữủ ự t ỗ ỗ ✤❛ ❣✐→❝ ✤ì♥✱ α✲✤❛ ❣✐→❝✱ ❜❛♦ α✲❧ã♠✳ ✲ P❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉ ❧➔ ❝→❝ t➼♥❤ ❝❤➜t ✈➔ ♠è✐ q✉❛♥ ❤➺ ❣✐ú❛ ❝→❝ ❦❤→✐ ♥✐➺♠ tr➯♥❀ ♠ët sè ✈➼ ❞ö ♠✐♥❤ ❤å❛ ✈➔ t➼♥❤ t♦→♥ ❝→❝ ✤è✐ t÷đ♥❣ ✤â✳ ✻✳ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉ ✲ ❉ị♥❣ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉ tr♦♥❣ ❤➻♥❤ ❤å❝ ✲ tỉ♣ỉ✳ ✲ ❙û ❞ư♥❣ ❝→❝ ♣❤÷ì♥❣ ự ỵ tt ự t ✼✳ ◆❤ú♥❣ ✤â♥❣ ❣â♣ ♠ỵ✐ ❝õ❛ ✤➲ t➔✐ ✲ ◆➯✉ r❛ ❝→❝ ♥❤➟♥ ①➨t ✈➲ ❜❛♦ α✲❧ã♠ ✲ ◆➯✉ r❛ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❜❛♦ ✭◆❤➟♥ ①➨t ✷✳✶✳✷✱ ✷✳✶✳✸✮✳ α✲❧ã♠ ✭❚➼♥❤ ❝❤➜t ✷✳✷✳✶ ✲ ✷✳✷✳✸✮ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t ❚➼♥❤ ❝❤➜t ✷✳✷✳✸ ♠➔ t➔✐ ❧✐➺✉ ❬✶❪ ❝❤➾ ❝❤ù♥❣ ♠✐♥❤ ❧÷đ❝✳ ✲ ❚r➻♥❤ ❜➔② ❝❤✐ t✐➳t ❝❤ù♥❣ ♠✐♥❤ ❝õ❛ ❚❤✉➟t t♦→♥ ✷✳✸✳✶ ✈➔ ❱➼ ❞ö ✷✳✸✳✷ ♠➔ t➔✐ ❧✐➺✉ ❬✶❪ ❦❤æ♥❣ ❝❤ù♥❣ ♠✐♥❤✳ ✲ ❚r➻♥❤ ❜➔② ❝❤✐ t✐➳t ❝❤ù♥❣ ♠✐♥❤ ❇ê ✤➲ ✷✳✸✳✼ ♠➔ t➔✐ ❧✐➺✉ ❬✶❪ ❝❤➾ ❝❤ù♥❣ ♠✐♥❤ ❧÷đ❝✳ ✲ ❚r➻♥❤ ❜➔② ❝❤✐ t✐➳t ❚❤✉➟t t♦→♥ ✷✳✸✳✽ ♠➔ t➔✐ ❧✐➺✉ ❬✶❪ ❝❤➾ ✤➲ ❝➟♣ ♥❣➢♥ ❣å♥✳ ❈❤÷ì♥❣ ✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ❚r♦♥❣ ữỡ ú tổ ợ t ởt số tự ❧➔♠ ❝ì sð ❝❤♦ ✈✐➺❝ tr➻♥❤ ❜➔② ♥ë✐ ❞✉♥❣ ❝❤➼♥❤ ỗ r ú tổ tr t ỗ tr ổ ❣✐❛♥ Rn ✈➔ ♠ët sè ✈➼ ❞ö ♠✐♥❤ ❤å❛ tr♦♥❣ ổ t r Rn t ỵ [p, q]✱ (p, q)✱ (p, q] ❧➛♥ ❧÷đt ❧➔ ✤♦↕♥ t❤➥♥❣ ✭❤❛② ❦❤♦↔♥❣ ✤â♥❣✮✱ ❦❤♦↔♥❣ ♠ð ✈➔ ♥û❛ ❦❤♦↔♥❣ ♥è✐ ❤❛✐ ✤✐➸♠ p ✈➔ q✳ ❚ù❝ ❧➔ [p, q] = {x ∈ Rn : λp + (1 − l)q : ≤ λ ≤ 1} (p, q) = {x ∈ Rn : λp + (1 − l)q : < λ < 1} (p, q] = {x ∈ Rn : λp + (1 − l)q : < λ ≤ 1} [p, q) = {x ∈ Rn : λp + (1 − l)q : ≤ λ < 1} p = q ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t✱ ♥➳✉ t❤➻ [p, q] = {p} ❝á♥ [p, q) = (p, q] = (p, q) = ∅✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ❬✽❪ ▼ët t➟♣ A Rn ữủ ỗ ợ x, y ∈ A t❤➻ [x, y] ⊂ A✳ ▼ët ❝→❝❤ t÷ì♥❣ ✤÷ì♥❣✿ ❚➟♣ t❤ü❝ α✿ ≤ α ≤ ❚➟♣ t A ỗ ợ (1 )x + y A ữủ t ỗ x, y tở A số ỗ r R ổ ỗ t rộ t t ỷ ✤÷í♥❣ t❤➥♥❣✱ ✈➔ t♦➔♥ ❜ë ✤÷í♥❣ t❤➥♥❣ t❤ü❝ ❧➔ ❝→❝ t ỗ r R2 t D ❧➔ ♠✐➲♥ ♥❣❤✐➺♠ ❝õ❛ ❤➺ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥   a1 x + b y ≤ c     a x+b y ≤c 2      a x+b y ≤c m D ❑❤✐ ✤â m (∗) m ❧➔ t ỗ ự D t rộ t❤➻ ❤✐➸♥ ♥❤✐➯♥ ◆➳✉ t➟♣ D ❦❤→❝ ré♥❣✳ ▲➜② t❤✉ë❝ t ỗ t p = (p1 , p2 ), q = (q1 , q2 ) ❜➜t ❦➻ t❤✉ë❝ D ✈➔ sè t❤ü❝ [0, 1]✳ ❑❤✐ ✤â✿ ✣➦t✿ D λp + (1 − λ)q ∈ D✱ t❤➟t ✈➟②✿ λp + (1 − λ)q = (λp1 + (1 − λ)q1 , λp2 + (1 − λ)q2 ) = (u, v) ❚❛ ❝â✿ (p1 , p2 ), (q1 , q2 ) ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❤➺ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ a1 p1 + b1 p2 ≤ c1 a1 q1 + b1 q2 ≤ c1 ❚ø ✤â✿ a1 u + b1 v = a1 (λp1 + (1 − λ)q1 ) + b1 (λp2 + (1 − λ)q2 ) = λ (a1 p1 + b1 p2 ) + (1 − λ) (a1 q1 + b1 q2 ) ≤ λc1 + (1 − λ)c1 = c1 ✳ (∗) ♥➯♥ t❛ ❝â✿ λ ✽ ❚❛ s✉② r❛ (u, v) (u, v) ❍♦➔♥ t♦➔♥ t÷ì♥❣ tü t❛ ❝ô♥❣ ❝â x + b i y ≤ c i ✱ (u, v) ❉♦ ✤â ♠å✐ p, q ❱➟② D a1 x + b y ≤ c tọ t ữỡ tr ợ tọ ♠➣♥ ❝→❝ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ i = 2; 3; ; m ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❤➺ ❜➙t ♣❤÷ì♥❣ tr➻♥❤ D t❤✉ë❝ ✈➔ ✈ỵ✐ ♠å✐ sè t❤ü❝ λ t❤✉ë❝ (∗)✱ ❤❛② λp + (1 − λ)q ∈ D ✈ỵ✐ [0; 1] t ỗ p, q ❧➔ ❤❛✐ ✤✐➸♠ ♣❤➙♥ ❜✐➺t tr♦♥❣ Rn ✳ ❑❤✐ õ t [p, q] ởt t ỗ ự ♠✐♥❤ ✳ ▲➜② ❑❤✐ ✤â✿ ❉♦ x, y ❜➜t ❦➻ t❤✉ë❝ [p, q] λx + (1 − λ)y ∈ [p; q]✱ x ∈ [p, q] ♥➯♥ ✈➔ sè t❤ü❝ λ t❤✉ë❝ [0; 1]✳ t❤➟t ✈➟②✿ x = λ p + (1 − λ ) q : ≤ λ ≤ 1✱ y = λ p + (1 − λ ) q : ≤ λ ≤ 1✳ ❚❛ s✉② r❛✿ λx + (1 − λ)y = λ [λ p + (1 − λ ) q] + (1 − λ) [λ p + (1 − λ ) q] = [λλ + (1 − λ)λ ] p + [λ(1 − λ ) + (1 − λ)(1 − λ )] q = Ap + Bq ✳ ≤ A, B ≤ ❉➵ t❤➜② A + B = ❉♦ ✤â λx + (1 − λ)y ∈ [p; q]✱ t❤✉ë❝ ✈ỵ✐ ♠å✐ x, y t❤✉ë❝ [p, q] ✈➔ ✈ỵ✐ ♠å✐ sè t❤ü❝ λ [0; 1]✳ ❱➟② ✤♦↕♥ t❤➥♥❣ [p, q] tr♦♥❣ Rn ❧➔ ởt t ỗ ỗ r ú tổ tr ỗ t t ỗ tr ổ Rn ❈❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t ❝→❝ t➼♥❤ ❝❤➜t ♥➔② ✈➔ ♠ët số ỗ tr ổ ❣✐❛♥ t❤➜♣ ❝❤✐➲✉ ✤÷đ❝ ❝❤➾ r❛✳ ✶✾ ❉♦ ✤â✱ ❣✐↔ sû ✤➣ s❛✐✱ ♥❣❤➽❛ ❧➔ ❦➸ tø ❦❤✐ ❜➢t ✤➛✉ t↕✐ C2 a ❚✐➳♣ ✤➳♥ t❛ ❝❤ù♥❣ tä✱ ✤✐➸♠ ❣✐❛♦ ♥❤❛✉ ♣❤↔✐ ♥➡♠ tr➯♥ ❝→❝ ✤♦↕♥ C1 x2,1 ❜ä ✤✐ t❤➻ x2,1 ❝➢t A1 , A2 ❚ø ✤â ●å✐ αi ✈➔ C1 ❧➔ ❣â❝ tr♦♥❣ t↕✐ ✤➾♥❤ ❝õ❛ ✤❛ ❣✐→❝ ax2,1 A2 ✳ ▲↕✐ ❝â✿ ❈→❝ ✤✐➸♠ x1,m A1 α− ❧ã♠ ❝õ❛ ✤❛ ❣✐→❝ x2,1 A1 ✈➔ ✈➔ ax2,1 x2,1 x2,2 x2,p b, A2 ✳ C1 = ✣✐➲✉ ♥➔② ♠➙✉ S✳ s➩ ♥➡♠ tr➯♥ ❝→❝ ✤♦↕♥ ✈➔ s ợ ữớ ú ổ tở C2 C1 A1 A3 ✈➔ βi x1,m b ✈➔ ax2,1 ❧➔ ❣â❝ tr♦♥❣ t↕✐ ✤➾♥❤ ❜➡♥❣ ❝→❝❤ ❧♦↕✐ ❜ä ❝→❝ ❝↕♥❤ ợ ữủ A3 x2,i x1,m b ❝â ❞✐➺♥ t➼❝❤ A2 ✳ a, b ♥➡♠ tr➯♥ ✤÷í♥❣ ỗ õ tr t x1,m ❝õ❛ A3 ♥❤ä ❤ì♥ ❝õ❛ A3 ✤➲✉ ❦❤ỉ♥❣ q✉→ ✈➔ ✈➔ ❧ã♠ ❈❤ó♥❣ t❛ ①➙② ❞ü♥❣ ✤❛ ❣✐→❝ ✈➔ ♥è✐ ❤❛✐ ✤➾♥❤ A1 x2,1 x1,i ♥❤ä ❤ì♥ ❞✐➺♥ t➼❝❤ ❝õ❛ ✤❛ ❣✐→❝ ❜➨ ❤ì♥ α− ❝❤ù❛ t➜t ❝↔ ❝→❝ ✤✐➸♠ ❝õ❛ ❉♦ ✤â ✤✐➸♠ ❣✐❛♦ ♥❤❛✉ ❝õ❛ x1,m b t↕✐ ởt tr ữớ ú ợ ax1,1 x1,2 z x1,m b✳ t❤✉➝♥ ✈➻ C2 t❤➻ c✳ ✈÷đt q✉❛ ♥❤❛✉ ➼t ♥❤➜t ♠ët ❧➛♥ t↕✐ ♠ët ✤✐➸♠ ❚❤➟t ✈➟②✱ ♥➳✉ b ✈➔ ❦➳t t❤ó❝ t↕✐ αm A2 ✳ ✈➔ ❣â❝ t↕✐ ✤➾♥❤ π+α ♥➯♥ x2,1 A3 ❧➔ ❝õ❛ α− A3 ♥❤ä ❤ì♥ β1 ✳ ❚ø ✤â ❝→❝ ❣â❝ tr♦♥❣ ✤❛ ❣✐→❝ ❝❤ù❛ t➟♣ ✤✐➸♠ ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ ✤à♥❤ ♥❣❤➽❛ ❜❛♦ α− S ❝â ❞✐➺♥ t➼❝❤ ❧ã♠✳ ❙✉② r❛ ❣✐↔ sû ✤➣ s❛✐✳ ❉♦ ✤â DA1 = DA2 ✳ ❱➟② t❛ ❝â A1 ✈➔ A2 ❝â t➟♣ ❝→❝ ✤➾♥❤ trò♥❣ ♥❤❛✉✳ ✷✳✸ ❱➲ ❜➔✐ t♦→♥ t➼♥❤ ❜❛♦ α✲❧ã♠ ✷✳✸✳✶ ❱➲ ❜➔✐ t♦→♥ t➼♥❤ α− ✤❛ ❣✐→❝ ❳➨t ❜➔✐ t♦→♥ s❛✉ ✤➙②✿ ❇➔✐ t♦→♥ ✶✿ ❚➻♠ ♠ët α− ✤❛ ❣✐→❝ ❝â t trữợ ởt t ỳ (0 < < ) ự trữợ ✤➼❝❤ ❧➔ t➻♠ ♠ët α− S ❧➔ t➟♣ ❤ú✉ ❤↕♥ ✤✐➸♠✱ ✤❛ ❣✐→❝ ❝❤ù❛ S x∈R ✈ỵ✐ ❞✐➺♥ t➼❝❤ ❧➔ ✈➔ α ❧➔ ❣â❝ ❝❤♦ x✳ ❘➜t ❦❤â ✤➸ ✤÷❛ r❛ ♠ët ❧í✐ ❣✐↔✐ ❝❤➼♥❤ ①→❝ ❝❤♦ ❜➔✐ t♦→♥ ♥➔② tr trữớ ủ tờ qt ợ S t t ❦ý tr♦♥❣ ♠➦t ♣❤➥♥❣ ✭sè ❧÷đ♥❣ ✤✐➸♠ ❝â t❤➸ r➜t ợ ọ trữợ t t r t tr➯♥ ❝â ❧✉ỉ♥ ❣✐↔✐ ✤÷đ❝ ❤❛② ❦❤ỉ♥❣✱ tù❝ ❧➔ ❝â tỗ t tt t ổ ữủ t tr♦♥❣ t❤í✐ ❣✐❛♥ ❤ú✉ ❤↕♥ ❤❛② ❦❤ỉ♥❣❄ ✷✵ ✣➸ ❝❤➾ r❛ ✤÷đ❝ ❝➙✉ tr↔ ❧í✐ ❧➔ ✧❝â✧✱ ❝❤ó♥❣ tỉ✐ ①➨t ❜➔✐ t♦→♥ t❤ù ❤❛✐ s❛✉ ✤➙②✱ ❝â t➯♥ ❣å✐ ❧➔ ✧❜➔✐ t♦→♥ tê♥❣ t➟♣ ❝♦♥✧ ✲ ♠ët ❜➔✐ t♦→♥ ♥ê✐ t✐➳♥❣ ✈➲ t➟♣ ❤ñ♣✿ ❇➔✐ t♦→♥ ✷✿ ●å✐ M = {m1, m2, , mn} ❧➔ t➟♣ ❝→❝ số tỹ k N trữợ ♠ët t➟♣ ❝♦♥ M M ❝õ❛ ♠➔ tê♥❣ ❝→❝ ♣❤➛♥ tû ❝õ❛ ♥â ❜➡♥❣ k✳ ✣➙② ❝ô♥❣ ❧➔ ♠ët ❜➔✐ t♦→♥ ❦❤ỉ♥❣ ❞➵ ✤➸ ✤÷❛ r❛ ❧í✐ ❣✐↔✐ ✭t❤✉➟t t♦→♥✮ ❝❤✉♥❣ ❝❤♦ ♠å✐ tr÷í♥❣ ❤đ♣ ❜➜t ❦ý✳ ❚➔✐ ❧✐➺✉ ❬✶✵❪ ✤➣ ❝❤➾ r❛ r➡♥❣ ❜➔✐ t♦→♥ ✷ ❧➔ ❧✉æ♥ ❣✐↔✐ ✤÷đ❝ tr♦♥❣ t❤í✐ ❣✐❛♥ ❤ú✉ ❤↕♥✳ ❚r♦♥❣ ♠ư❝ ♥➔②✱ ❝❤ó♥❣ tæ✐ tr➻♥❤ ❜➔② t❤✉➟t t♦→♥ ✧r❡❞✉❝❡✧ tø ❜➔✐ t♦→♥ ✷ s❛♥❣ ❜➔✐ t♦→♥ ✶✱ tù❝ ❧➔ ❝❤✉②➸♥ ✤ê✐ tø ✈✐➺❝ ❣✐↔✐ ❜➔✐ t♦→♥ ✷ s❛♥❣ ✈✐➺❝ ❣✐↔✐ ❜➔✐ t♦→♥ ✶✱ tø ✤â ❝❤➾ r❛ r➡♥❣ ❜➔✐ t♦→♥ t➼♥❤ ❞✐➺♥ t➼❝❤ trữợ < < ♠ët t➟♣ ✤✐➸♠ ✈ỵ✐ ❧➔ ❧✉ỉ♥ ❝â t❤✉➟t t♦→♥ ❣✐↔✐ ✤÷đ❝ tr♦♥❣ t❤í✐ ❣✐❛♥ ❤ú✉ ❤↕♥✳ ❚❤✉➟t t♦→♥ ✷✳✸✳✶✳ ✭❬✶❪✮❚❤✉➟t t♦→♥ ❝❤✉②➸♥ ✤ê✐ ❜➔✐ t♦→♥ ✷ s❛♥❣ ❜➔✐ t♦→♥ ✶✳ ✭❚❤✉➟t t♦→♥ tr➻♥❤ ❜➔② ❝→❝❤ ❣✐↔✐ ❜➔✐ t♦→♥ ✷ t❤æ♥❣ q✉❛ ❜➔✐ t♦→♥ ✶✮ ■♥♣✉t✿ M = {m1 , m2 , , mn } ❖✉t♣✉t✿ ❚➟♣ ❝♦♥ M ❝õ❛ M ❧➔ t➟♣ ❝→❝ sè tü ♥❤✐➯♥ ✈➔ ♠➔ tê♥❣ ❝→❝ ♣❤➛♥ tû ❝õ❛ ❚r♦♥❣ t❤✉➟t t♦→♥ ♥➔②✱ t❛ s➩ ①➙② ❞ü♥❣ ♠ët ❣✐→❝ ✤➲✉ R ❝↕♥❤ M trữợ k õ t x tø n− t✱ ❜➡♥❣ ❝→❝❤ t❤➯♠ ✈➔♦ ♠✐➲♥ tr♦♥❣ n− ❣✐→❝ ✤➲✉ ✤â ❝→❝ ✤✐➸♠ tr➯♥ ✤÷í♥❣ tr✉♥❣ trü❝ ộ ởt ú ữợ t r = max {mi } ữợ t t = 2r cot ữợ R ữợ E = {e1 , e2 , , en } mi 1in n ữợ α ✳ ✲ ✤❛ ❣✐→❝ ✤➲✉ s❛♦ ❝❤♦ ✤ë ❞➔✐ ♠é✐ ❝↕♥❤ t✳ ❧➔ t➟♣ t➜t ❝↔ ❝→❝ ❝↕♥❤ ❝õ❛ R✳ n ✤✐➸♠ Q = {q1 , q2 , , qn } ✈➔♦ tr♦♥❣ R s❛♦ ❝❤♦ ♠é✐ ✤✐➸♠ qi ♥➡♠ tr➯♥ ✤÷í♥❣ tr✉♥❣ trü❝ ❝õ❛ qi tữỡ ự ữợ t kN ei ei ♠ët ❦❤♦↔♥❣ ❝→❝❤ mi ✳ ▼é✐ ✤✐➸♠ mi ✳ x = AR − kt ✈➔ S = Q ∪ VR s❛♦ ❝❤♦ AR = nt2 π cot n ❧➔ ❞✐➺♥ ✷✶ R t➼❝❤ ❝õ❛ ❝→❝ ❣✐→ trà ✈➔ mi ữợ ữợ t VR tr R ❧➔ ❝→❝ ✤➾♥❤ ❝õ❛ M Q ❧➔ t➟♣ ❝→❝ qi tữỡ ự ợ tr M ⊂M ✈➔ S ✈ỵ✐ ❞✐➺♥ t➼❝❤ x ✭❇➔✐ t♦→♥ ✶✮✳ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥✳ ▼é✐ ✤✐➸♠ ✤❛ ❣✐→❝ ✤➣ t tữỡ ự ợ mi M qi tr ❝õ❛ α− ✳ ❈❤ù♥❣ ♠✐♥❤ ✳ ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ ữợ tr s r M t♦→♥ ✷✳ Aj H mi αi Ai I t ●✐↔ sû tù♥❣ ❝↕♥❤ n− ei t❤✉ ✤÷đ❝ ❧➔ ❣✐→❝ ✤➲✉ ❝â t➙♠ ❧➔ ✈➔ ❣✐→ trà αi ✳ mi ✱ ✈➔ ❝â t ✤➾♥❤ ❧✐➯♥ t✐➳♣ ❧➔ Ai , Aj ei ❝õ❛ ✤➦t t÷ì♥❣ α− ✤❛ ❣✐→❝ tèt ♥❤➜t ✤➸ ♠é✐ ❣â❝ tr♦♥❣ ❝õ❛ ✤❛ ❣✐→❝ t❤✉ π + α✳ r = max {mi } 1in ữợ ✣➦t t = 2r cot ❚❛ ❝â✿ Ai Aj = 2AH 2π − αi = 2mi tan 2π − (α + π) ≥ 2mi tan π−α = 2mi tan = 2mi cot ữợ ✶✳ ✣➦t ❣â❝ tr♦♥❣ t↕✐ ✤➾♥❤ t÷ì♥❣ ù♥❣ ❝↕♥❤ ❚❛ ❝➛♥ t➻♠ sè t❤ü❝ ✤÷đ❝ ✤➲✉ ❦❤ỉ♥❣ q✉→ I ✷✷ α , ∀i = 1, n✳ α α ❚❛ ❧➜② t = max {mi } cot = 2r cot 1in 2 ữợ R n ✲ ✤❛ ❣✐→❝ ✤➲✉ s❛♦ ❝❤♦ ✤ë ❚ø ✤â t 2mi cot ữợ E = {e1 , e2 , , en } ữợ ✺✳ ❚❤➯♠ ✈➔♦ n ✤✐➸♠ ❞➔✐ ♠é✐ ❝↕♥❤ t✳ ❧➔ t➟♣ t➜t ❝↔ ❝→❝ ❝↕♥❤ ❝õ❛ Q = {q1 , q2 , , qn } ♥➡♠ tr➯♥ ✤÷í♥❣ tr✉♥❣ trü❝ ❝õ❛ ù♥❣ ✳ ei ❝→❝❤ ei ✈➔♦ tr♦♥❣ ♠ët ❦❤♦↔♥❣ ❝→❝❤ R✳ R s❛♦ ❝❤♦ ♠é✐ ✤✐➸♠ mi ✳ ▼é✐ ✤✐➸♠ qi qi t÷ì♥❣ mi ✳ kt nt2 ữợ t x = AR S = Q ∪ VR s❛♦ ❝❤♦ AR = cot ❧➔ ❞✐➺♥ t➼❝❤ n ❝õ❛ R ✈➔ VR ❧➔ ❝→❝ ✤➾♥❤ ❝õ❛ R ✈➔ Q ❧➔ t➟♣ ❝→❝ qi tữỡ ự ợ tr mi tr M ✳ ●å✐ pR ✈➔ rR ✤➲✉ R✳ ❑❤✐ ✤â ❞✐➺♥ t➼❝❤ ❝õ❛ ❧➛♥ ❧÷đt ❧➔ ♥û❛ ❝❤✉ ✈✐ ✈➔ ❜→♥ ❦➼♥❤ ✤÷í♥❣ trá♥ ♥ë✐ t✐➳♣ ❝õ❛ n− ❣✐→❝ ✤➲✉ R n− ❣✐→❝ ❧➔✿ nt t π nt2 π AR = pR rR = cot = cot 2 n n ❑❤✐ ✤â ❞✐➺♥ t➼❝❤ α− ✤❛ ❣✐→❝ ①➙② ❞ü♥❣ ✤÷đ❝ ❧➔✿ π kt kt nt2 = cot − x = AR − n ữợ ữợ ộ ✤â M ⊂M ✤❛ ❣✐→❝ tr➯♥ qi S ✈ỵ✐ ❞✐➺♥ t➼❝❤ tr➯♥ ❜✐➯♥ ❝õ❛ α− x✳ ✤❛ ❣✐→❝ ✤➣ t➼♥❤ tữỡ ự ợ mi M ❜➔✐ t♦→♥✳ ◆❤÷ ✈➟②✱ ❚❤✉➟t t♦→♥ ✷✳✸✳✶ ✤➣ ❣✐↔✐ ❜➔✐ t♦→♥ ✷ ✭tê♥❣ t➟♣ ❝♦♥✮ t❤æ♥❣ q✉❛ ❜➔✐ t♦→♥ ✶ ữợ t ữợ tt t♦→♥ ♥➔② ✤➲✉ sû ❞ư♥❣ ❝→❝ t➼♥❤ t♦→♥ ❝➜♣ ✭t➼♥❤ t♦→♥ tr♦♥❣ t❤í✐ ❣✐❛♥ ✤❛ t❤ù❝✮✳ ❍ì♥ ♥ú❛✱ t➔✐ ❧✐➺✉ ❬✶✵❪ ✤➣ ❝❤➾ r❛ r➡♥❣ ❜➔✐ t♦→♥ ✷ ❧➔ ổ ữủ tr tớ ỳ ỵ t❤✉②➳t tr♦♥❣ ♥❣➔♥❤ ❦❤♦❛ ❤å❝ ♠→② t➼♥❤✱ ❜➔✐ t♦→♥ ✷ ✭✤➣ ❜✐➳t✮ ✤➣ ✤÷đ❝ ✧r❡❞✉❝❡✧ ✈➲ ❜➔✐ t♦→♥ ✶ ✭❝❤÷❛ ❜✐➳t ❣✐↔✐✮✳ ❚❛ ❝â ♥❤➟♥ ①➨t s❛✉ ✤➙②✳ ◆❤➟♥ ①➨t✿ ỗ t tt t ữủ t t t➼❝❤ ❦❤æ♥❣ ✤ê✐ ❝õ❛ ♠ët t➟♣ ❤ú✉ ❤↕♥ ✤✐➸♠✱ < α < π✮ α− ✤❛ ❣✐→❝ ❝â ❞✐➺♥ tr♦♥❣ t❤í✐ ❣✐❛♥ ❤ú✉ ❤↕♥✳ ✷✸ ❙❛✉ ✤➙② t❛ ✤÷❛ r❛ ♠ët ✈➼ ❞ö ♠✐♥❤ ❤å❛ ❝❤♦ t❤✉➟t t♦→♥ ✷✳✸✳✶✳ ❱➼ ❞ö ✷✳✸✳✷✳ α= π 2, ✈➔ ✭❬✶❪✮ ❈❤♦ t➟♣ ✈➔ sè tü ♥❤✐➯♥ 8✳ ❉♦ ✤â M = {m1 = 2; m2 = 3; m3 = 4; m4 = 6; m5 = 7; m6 = 8}✱ k = 14✳ ❑❤✐ ✤â M = {2, 4, 8} α− ✤❛ ❣✐→❝ ❝â ❞✐➺♥ t➼❝❤ x ✤✐ q✉❛ ❝→❝ ✤✐➸♠ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✷✳ r = max {mi } = max {2; 3; 4; 6; 7; 8} = 8✳ 1≤i≤6 π α = 2.8 cot = 16✳ ✣➦t t = 2r cot ●å✐ R ❧➔ ❧ö❝ ❣✐→❝ ✤➲✉ s❛♦ ❝❤♦ ✤ë ❞➔✐ ♠é✐ ❝↕♥❤ ❜➡♥❣ 16✳ ❈❤ù♥❣ ♠✐♥❤ ✳ ✣➦t ●å✐ E = {e1 , e2 , , en } ❚❤➯♠ ✈➔♦ ei ✤✐➸♠ ❧➔ t➟♣ t➜t ❝↔ ❝→❝ ❝↕♥❤ ❝õ❛ {q1 , q2 , , q6 } tr♦♥❣ R R✳ t÷ì♥❣ ù♥❣ ♥➡♠ tr➯♥ ✤÷í♥❣ tr✉♥❣ trü❝ mi ✳ √ kt 6.16 π 14.16 ✣➦t x = AR − = cot − = 384 − 112 ✈➔ S = Q ∪ VR ✈ỵ✐ 6.162 π AR = cot ❧➔ ❞✐➺♥ t➼❝❤ ❝õ❛ R ✈➔ VR ❧➔ ❝→❝ ✤➾♥❤ ❝õ❛ R ✈➔ Q ❧➔ t➟♣ qi tữỡ ự ợ trà mi tr♦♥❣ M ✳ ❝→❝ ❝↕♥❤ ❚➼♥❤ ✣➦t α− M s❛♦ ❝❤♦ ❦❤♦↔♥❣ ❝→❝❤ tỵ✐ ❝→❝ ❝↕♥❤ ✤❛ ❣✐→❝ ❝õ❛ t➟♣ ✤✐➸♠ S ❝â ❞✐➺♥ t➼❝❤ ei x✳ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❝♦♥✳ ▼é✐ ✤✐➸♠ t➼♥❤ t÷ì♥❣ ù♥❣ ✈ỵ✐ mi ∈ M ✳ ❉♦ ✤â ❍➻♥❤ tr→✐✿ ▲ö❝ ❣✐→❝ ✤➲✉ ❍➻♥❤ ♣❤↔✐✿ ❇❛♦ R ❧➔ qi tr➯♥ ❜✐➯♥ ❝õ❛ α− ✤❛ ❣✐→❝ ✤➣ M = {2; 4; 8}✳ ✈ỵ✐ ✤ë ❞➔✐ ♠é✐ ❝↕♥❤ ❜➡♥❣ ✶✻ ✈➔ ❞✐➺♥ t➼❝❤ ❧➔ 6.162 π AR = cot ✳ 14.16 α−❧ã♠ ✈ỵ✐ ❞✐➺♥ t➼❝❤ x = AR − ✳ ✷✹ ✷✳✸✳✷ ❱➲ ❜➔✐ t♦→♥ t➼♥❤ ❜❛♦ α−❧ã♠ ❳➨t ❜➔✐ t♦→♥ s❛✉ ✤➙②✿ ❇➔✐ t♦→♥ ✸✿ ❚➻♠ ❜❛♦ α✲❧ã♠ ❝õ❛ t➟♣ ❤ú✉ ❤↕♥ ✤✐➸♠ ✈ỵ✐ < α < π✳ ❚ù❝ ❧➔✿ ❈❤♦ S ❧➔ t➟♣ ❤ú✉ ❤↕♥ ✤✐➸♠ tr♦♥❣ ♠➦t ♣❤➥♥❣✱ t➼♥❤ ❜❛♦ ❚÷ì♥❣ tü ♥❤÷ ❜➔✐ t♦→♥ t➼♥❤ α−✤❛ α✲❧ã♠ ❝õ❛ ❣✐→❝✱ ❜➔✐ t♦→♥ t➼♥❤ ❜❛♦ S ✈ỵ✐ < α < π✳ α−❧ã♠ ❝ơ♥❣ ❧➔ ❜➔✐ t♦→♥ r➜t ❦❤â ✤➸ ✤÷❛ r❛ t❤✉➟t t♦→♥ ❝❤➼♥❤ ①→❝ ❣✐↔✐ ✤÷đ❝ ❝❤♦ ♠å✐ tr÷í♥❣ ❤đ♣ ❜➜t ❦ý S ❝õ❛ tr t ọ trữợ t t r ❜➔✐ t♦→♥ ✸ ❝â ❧✉ỉ♥ ❣✐↔✐ ✤÷đ❝ ❤❛② ❦❤ỉ♥❣✱ tù❝ õ tỗ t tt t ổ ữủ ❜➔✐ t♦→♥ tr♦♥❣ t❤í✐ ❣✐❛♥ ❤ú✉ ❤↕♥ ❤❛② ❦❤ỉ♥❣❄ ✣➸ ❝❤➾ r❛ ✤÷đ❝ ❝➙✉ tr↔ ❧í✐ ❧➔ ✧❝â✧✱ ❝❤ó♥❣ tỉ✐ ①➨t ❜➔✐ t♦→♥ ❝â t➯♥ ❣å✐ ✧❜➔✐ t♦→♥ ♣❤➙♥ ♥❤â♠ t❛♠ ❣✐→❝✧ ✭tr÷í♥❣ ❤đ♣ r✐➯♥❣ ❝õ❛ ❜➔✐ t♦→♥ ♣❤➙♥ ♥❤â♠✮✱ ♠ët ❜➔✐ t♦→♥ ❦❤ỉ♥❣ ❞➵ ✤➸ ✤÷❛ r❛ ❧í✐ ❣✐↔✐ ✭t❤✉➟t t♦→♥✮ ❝❤✉♥❣ ❝❤♦ ♠å✐ tr÷í♥❣ ❤đ♣ ❜➜t ❦ý✳ ❚➔✐ ❧✐➺✉ ❬✷❪ ✈➔ ❬✶✵❪ ✤➣ ❝❤➾ r❛ r➡♥❣ ❜➔✐ t♦→♥ ♣❤➙♥ ♥❤â♠ t❛♠ ❣✐→❝ ❧➔ ❧✉ỉ♥ ❣✐↔✐ ✤÷đ❝ tr♦♥❣ t❤í✐ ❣✐❛♥ ❤ú✉ ❤↕♥✳ ❚r♦♥❣ ♠ư❝ ♥➔②✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② t❤✉➟t t♦→♥ ✧r❡❞✉❝❡✧ tø ❜➔✐ t♦→♥ ♣❤➙♥ ♥❤â♠ t❛♠ ❣✐→❝ s❛♥❣ ❜➔✐ t♦→♥ ✸✱ tù❝ ❧➔ ❝❤✉②➸♥ ✤ê✐ tø ✈✐➺❝ ❣✐↔✐ ❜➔✐ t♦→♥ ♣❤➙♥ ♥❤â♠ t❛♠ ❣✐→❝ s❛♥❣ ✈✐➺❝ ❣✐↔✐ ❜➔✐ t♦→♥ ✸✱ tø ✤â ❝❤➾ r❛ r➡♥❣ ❜➔✐ t♦→♥ t➼♥❤ ❜❛♦ α✲❧ã♠ ❝õ❛ t➟♣ ❤ú✉ ❤↕♥ ✤✐➸♠ ✭0 < α < π✮ ❧➔ ❧✉ỉ♥ ❝â t❤✉➟t t♦→♥ ❣✐↔✐ ✤÷đ❝ tr♦♥❣ t❤í✐ ❣✐❛♥ ❤ú✉ ❤↕♥✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✸✳✸✳ ❈❤♦ d : S × S → N ❧➔ ❤➔♠ ❦❤♦↔♥❣ ❝→❝❤ ✈ỵ✐ S k, B ✳ ❧➔ ♠ët t➟♣ ✤✐➸♠ ✈➔ ❤❛✐ sè tü ♥❤✐➯♥ ❇➔✐ t♦→♥ ♣❤➙♥ ♥❤â♠ ❧➔ ❝❤➾ r õ tỗ t ổ ởt t❤➔♥❤ t❤➻ ✭❬✶✵❪✮✭❇➔✐ t♦→♥ ♣❤➙♥ ♥❤â♠✮✳ k t➟♣ ❝♦♥ C1 , C2 , , Ck s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ i = 1; k t❤✉ë❝ Ci p1 , p2 , , pr ∈ S ✈➔ ✈➔ ✈ỵ✐ ♠å✐ x, y S d(x, y) ≤ B ✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✸✳✹✳ ❈❤♦ ∆a1 a2 a3 ✈➔ i = j ∈ {1; 2; 3} ✤❛ ❣✐→❝ ✤➾♥❤ pi S ✭❬✶❪✮ ✭α ✲ ✤÷í♥❣✮ ❧➔ ♠ët t➟♣ ✤✐➸♠ tr♦♥❣ t ợ t ữớ ú p1 p2 pn aj L = p p p n aj ❧➔ ♠ët α−✤÷í♥❣ ♥➳✉ ❧➔ ♠ët ✤❛ ❣✐→❝ ✤ì♥ ❦❤→❝ ré♥❣ s❛♦ ❝❤♦ ♠å✐ ❣â❝ tr♦♥❣ t↕✐ ❝→❝ tr♦♥❣ ✤❛ ❣✐→❝ ❧➔ ❧ỵ♥ ❤ì♥ α✳ ✷✺ ✣à♥❤ ♥❣❤➽❛ ✷✳✸✳✺✳ ❈❤♦ S ✭❬✶❪✮ ✭❇➔✐ t♦→♥ ♣❤➙♥ ♥❤â♠ t❛♠ ❣✐→❝✮ ❧➔ t➟♣ ✤✐➸♠ ♥➡♠ tr♦♥❣ ∆a1 a2 a3 ✳ ❇➔✐ t♦→♥ ♣❤➙♥ ♥❤â♠ t t r õ tỗ t ❤❛② ❦❤æ♥❣ ♠ët ♣❤➙♥ ❤♦↕❝❤ s❛♦ ❝❤♦ S t❤➔♥❤ ✸ ♣❤➛♥ C1 , C2 , C3 rí✐ ♥❤❛✉ ∀1 ≤ i ≤ ✈➔ ∀x, y ∈ Ci s❛♦ ❝❤♦ d (x, y) 1, ợ d : S ìS −→ {0; 1; 2} ❧➔ ❤➔♠ ❦❤♦↔♥❣ ❝→❝❤ ✤÷đ❝ ❝❤♦ ❜ð✐ ❝æ♥❣ t❤ù❝✿    ♥➳✉ x = y d (x, y) = tỗ t − ✤÷í♥❣ , , x, , y, , aj   ♥➳✉ ♥❣÷đ❝ ❧↕✐✳ ❙❛✉ ✤➙② t❛ ♥❤➢❝ ❧↕✐ ♥ë✐ ❞✉♥❣ ❝õ❛ ỵ P ữủ ự r r P ✭✶✽✺✾ ✕ ✶✾✹✸✮✳ ❇ê ✤➲ ✷✳✸✳✻✳ ✭✣à♥❤ ❧➼ P✐❝❦✮ ❈❤♦ P ❧➔ ♠ët ✤❛ ❣✐→❝ ✤ì♥ ❝â ❝→❝ ✤➾♥❤ ❧➔ ❝→❝ ✤✐➸♠ ♥❣✉②➯♥✱ t❤✉ë❝ ♠✐➲♥ tr♦♥❣ ❝õ❛ ❝â ❞✐➺♥ t➼❝❤ ❝õ❛ P P ✈➔ H I ❧➔ sè ✤✐➸♠ ♥❣✉②➯♥ ❧➔ sè ✤✐➸♠ ♥❣✉②➯♥ ♥➡♠ tr➯♥ ❜✐➯♥ ❝õ❛ P✳ ❑❤✐ ✤â t❛ ✤÷đ❝ t➼♥❤✿ AP = I + H − ❈❤ù♥❣ ♠✐♥❤ ✳ ❉ị♥❣ ♠ët ✤÷í♥❣ ❣➜♣ ❦❤ó❝ t❤➔♥❤ ❤❛✐ ✤❛ ❣✐→❝ ✤ì♥✿ ✣❛ ❣✐→❝ γ P1 ♥è✐ ❤❛✐ ✤➾♥❤ ❦❤æ♥❣ ❦➲ ♥❤❛✉ ❝õ❛ ✈➔ ✤❛ ❣✐→❝ ❚❛ t❤➜② ♥➳✉ ❝ỉ♥❣ t❤ù❝ P✐❝❦ ✤ó♥❣ ✈ỵ✐ ❣å✐ n ❧➔ sè ✤✐➸♠ ♥❣✉②➯♥ tr➯♥ γ P1 ✈➔ P ✤➸ ❝➢t P P2 ✳ P2 t❤➻ ♥â ❝ơ♥❣ ✤ó♥❣ ✈ỵ✐ P ✳ ❚❤➟t ✈➟②✱ t❛ ❝â I = IP1 + IP2 + n − 2, H = HP1 + HP2 − 2n + 2✱ s✉② r❛ 1 AP = AP1 + AP2 = IP1 + HP1 − + IP2 + HP2 − = I + H − 2 ❱➟② ❝ỉ♥❣ t❤ù❝ P✐❝❦ ✤ó♥❣ ợ P ỡ ỳ ự t tữỡ tü✱ ♥➳✉ ❝ỉ♥❣ t❤ù❝ P✐❝❦ ✤ó♥❣ ✈ỵ✐ ❤❛✐ tr♦♥❣ ❜❛ ✤❛ ❣✐→❝ P ✱ P2 ✱ P t❤➻ ♥â ❝ơ♥❣ ✤ó♥❣ ✈ỵ✐ ✤❛ ❣✐→❝ ❝á♥ ❧↕✐✳ ❉♦ ✤â t❛ ❝❤➾ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ ❝ỉ♥❣ t❤ù❝ P✐❝❦ ✤ó♥❣ ❝❤♦ ❝→❝ t❛♠ ❣✐→❝ ❧➔ ①♦♥❣✳ ✷✻ ❚❛ ❝â✱ ✈ỵ✐ ♠ët t❛♠ ❣✐→❝ ❜➜t ❦➻ tr♦♥❣ R2 ❝â ✤➾♥❤ (x1 , y1 ), (x1 , y1 ), (x1 , y1 ) ❧✉ỉ♥ ❜à ❝❤ù❛ tr♦♥❣ ❤➻♥❤ ❝❤ú ♥❤➟t ❣✐ỵ✐ ❤↕♥ ❜ð✐ ❝→❝ ✤÷í♥❣ t❤➥♥❣ x = min{x1 , x2 , x3 }, x = max{x1 , x2 , x3 } y = min{y1 , y2 , y3 }, y = max{y1 , y2 , y3 } ❚ø ✤â✱ ♠ët t❛♠ ❣✐→❝ ❜➜t ❦➻✱ t❛ ❧✉æ♥ ❝â t❤➸ ❣❤➨♣ t❤➯♠ ✈➔♦ ❝→❝ t❛♠ ❣✐→❝ ✈✉æ♥❣ ✈➔ ❝→❝ ❤➻♥❤ ❝❤ú ♥❤➟t ❝â ❝→❝ õ ổ ũ ữỡ ợ trử tồ ✤➸ t↕♦ t❤➔♥❤ ♠ët ❤➻♥❤ ❝❤ú ♥❤➟t ❝â ❝→❝ ❝↕♥❤ ũ ữỡ ợ trử tồ s t ❤✐➺♥ ✤✐➲✉ ✤â✮✳ a1 a1 a2 a3 a2 a3 a1 a1 a3 a2 a3 a2 ❉♦ ✤â✱ t❛ ❝❤➾ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ ❝ỉ♥❣ t❤ù❝ P✐❝❦ ✤ó♥❣ ❝❤♦ ❝→❝ t❛♠ ❣✐→❝ ổ õ õ ổ ũ ữỡ ợ trö❝ tå❛ ✤ë ✈➔ ❝→❝ ❤➻♥❤ ❝❤ú ♥❤➟t ❝â ❝→❝ ũ ữỡ ợ trử tồ ỵ P ữủ sỷ tr ❝❤ù♥❣ ♠✐♥❤ ♣❤➛♥ ✤↔♦ ❝õ❛ ❜ê ✤➲ s❛✉ ✤➙②✳ ❇ê ✤➲ ✷✳✸✳✼✳ ✭❬✶❪✮❈❤♦ S ❧➔ t➟♣ ✤✐➸♠ ♥➡♠ tr♦♥❣ ∆a1a2a3✳ ✈➔ A ❧➔ ♠ët ❜❛♦ α ✲ ❧ã♠ ❝õ❛ t➟♣ S ∪ {a1 , a2 , a3 }✳ A ✤✐ q✉❛ ♠å✐ ✤✐➸♠ ❝õ❛ S ♥❤â♠ t❛♠ ❣✐→❝ ✈ỵ✐ ❤➔♠ ❦❤♦↔♥❣ ❝→❝❤ ❝❤✐❛ S t❤➔♥❤ ✸ ♣❤➛♥✳ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ❜➔✐ t♦→♥ ♣❤➙♥ dβ : S × S −→ {0; 1; 2} ✭✈ỵ✐ β = π − α✮ ✤➣ ✷✼ ❈❤ù♥❣ ♠✐♥❤ ✳ a1 u1,j a1,j a3 a2 ✯✮ P❤➛♥ t❤✉➟♥✿ ●✐↔ sû ◆➳✉ A S = {x1 , x2 , , xn }✳ ✤✐ q✉❛ ❝→❝ ✤✐➸♠ ❝õ❛ a1 ❳➨t ✤÷í♥❣ ♥è✐ ✣❛ ❣✐→❝ a2 ✈➔ ✤➲✉ ❧ỵ♥ ❤ì♥ ✤➾♥❤ x1j β✱ t❤➻ t➟♣ ❤đ♣ ❝→❝ ✤➾♥❤ ❝õ❛ ❜ð✐ ❜✐➯♥ ❝õ❛ a1 x11 x12 x1m a2 1; m S A✱ ❣✐↔ sû ❧➔ A ❧➔ a1 x11 x12 x1m a2 t❤ä❛ ♠➣♥✿ ❧➔ ✤❛ ❣✐→❝ ✤ì♥ ❝â ❝→❝ ❣â❝ tr♦♥❣ t↕✐ ❝→❝ ✤➾♥❤ ✈➻ ♥➳✉ t❛ ❣å✐ v1j , ∀j = 1; m x1j , ∀j = ❧➔ sè ✤♦ ❣â❝ tr♦♥❣ t÷ì♥❣ ù♥❣ t↕✐ t❤➻ x1j = 2π − u1j > 2π − (π + α) = 2π − (π + (π − β)) = β ✱ t↕✐ ✤➾♥❤ S ∪ {a1 , a2 , a3 }✳ x1j õ ữớ ợ u1j õ tr A a1 x11 x12 x1m a2 ❧➔ β− ✤÷í♥❣✳ ❍♦➔♥ t♦➔♥ t÷ì♥❣ tü✿ ✣÷í♥❣ ♥è✐ ✈➔ a1 a2 ✈➔ a3 ❜ð✐ ❜✐➯♥ ❝õ❛ ❜ð✐ ❜✐➯♥ ❝õ❛ A✱ ❣✐↔ sû ❧➔ ❚ø ✤â t❛ ❝â ♣❤➙♥ ❤♦↕❝❤ A✱ a3 x31 x32 x3p a1 ❝õ❛ S t❤➔♥❤ ✸ ♣❤➛♥ t❤➻ ●✐↔ sû r➡♥❣ dβ ✳ ❑❤✐ ✤â✱ S A β− dβ : S × S −→ {0; 1; 2} ✤✐ q✉❛ ♠å✐ ✤✐➸♠ ❝õ❛ ✤÷đ❝ ♣❤➙♥ ❤♦↕❝❤ t❤➔♥❤ C1 aj , aj C2 ak , ak C3 ❉♦ ✤â✱ ✤❛ ❣✐→❝ ✤➲✉ ❧➔ ❝→❝ ✈➔ ✤÷í♥❣ ♥è✐ a3 ✤÷í♥❣✳ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ♣❤➙♥ ♥❤â♠ t❛♠ ❣✐→❝✳ ✯✮ P❤➛♥ ✤↔♦✿ ◆➳✉ ❤➔♠ ❦❤♦↔♥❣ ❝→❝❤ S a2 x21 x22 x2n a3 {C1 = {x11 , x12 , , x1m } , C2 = {x21 , x22 , , x2n } , C3 = {x31 , x32 , , x3p }} ❝❤✐❛ ❣✐↔ sû ❧➔ A = C aj C ak C ✭✈ỵ✐ ♣❤➛♥ S✳ α− β = π − α✮ ✤➣ ❚❤➟t ✈➟②✱ C1 , C2 , C3 ❜ð✐ ❤➔♠ ❦❤♦↔♥❣ ❝→❝❤ i = j = k ∈ {1; 2; 3} ởt ợ ữớ ✤❛ ❣✐→❝ ✤✐ q✉❛ ❝→❝ ✤✐➸♠ ❝õ❛ S✳ ✷✽ ❚❛ ❝❤✐❛ ❝→❝ ✤✐➸♠ tr➯♥ ❤➺ trö❝ tå❛ ✤ë ✤õ ❜➨ s❛♦ ❝❤♦ ❝→❝ ✤✐➸♠ ❝õ❛ S ✈➔ a1 , a2 , a3 ✤➲✉ ❧➔ ❝→❝ ✤✐➸♠ ♥❣✉②➯♥✳ α− ❧ã♠ ❝õ❛ S ∪ {a1 , a2 , a3 } ❦❤æ♥❣ ✤✐ q✉❛ ●✐↔ sû ♣❤↔♥ ❝❤ù♥❣ r➡♥❣ A2 ❧➔ ♠ët ❜❛♦ ❤➳t t➜t ❝↔ ❝→❝ ✤✐➸♠ ❝õ❛ S✳ ✣✐➲✉ ♥➔② ❞➝♥ ✤➳♥✱ sè ✤✐➸♠ ♥❣✉②➯♥ t❤✉ë❝ ♠✐➲♥ tr♦♥❣ ❝õ❛ A2 ♥❤✐➲✉ ❤ì♥ ❝õ❛ ✤â✱ t❤❡♦ ✤à♥❤ ❧➼ P✐❝❦✱ ❤ì♥ A2 ✳ A A ✈➔ sè ✤✐➸♠ ♥❣✉②➯♥ tr➯♥ ❜✐➯♥ ❝õ❛ ❧➔ α− ✤❛ ❣✐→❝ ❝❤ù❛ ✣✐➲✉ ♥➔② ❧➔ ♠➙✉ t❤✉➝♥ ✈➻ A2 ❧➔ ❜❛♦ A2 S ∪ {a1 , a2 , a3 } α− ❧ã♠ ❝õ❛ ➼t ❤ì♥ ❝õ❛ A✳ ❚ø ♠➔ ❝â ❞✐➺♥ t➼❝❤ ❜➨ S ∪ {a1 , a2 , a3 }✳ ❇ê ✤➲ tr➯♥ ✤÷đ❝ sû ❞ư♥❣ ✤➸ ✤÷❛ r❛ t❤✉➟t t♦→♥ s❛✉ ✤➙②✳ ❚❤✉➟t t♦→♥ ✷✳✸✳✽✳ ✭❬✶❪✮✭❚❤✉➟t t♦→♥ ❝❤✉②➸♥ ✤ê✐ ❜➔✐ t♦→♥ ♣❤➙♥ ♥❤â♠ t❛♠ ❣✐→❝ ✈➲ ❜➔✐ t♦→♥ ✸✮✳ ■♥♣✉t✿ ❈❤♦ S ❖✉t♣✉t✿ ❚➻♠ ❧➔ t➟♣ ✤✐➸♠ ♥➡♠ tr♦♥❣ C1 , C2 , C3 ♣❤➙♥ ❤♦↕❝❤ ∆a1 a2 a3 ✳ S t❤➔♥❤ ✸ ♣❤➛♥ rí✐ ♥❤❛✉ s❛♦ ❝❤♦ dα (x, y) ≤ 1, ✈ỵ✐ ∀x, y ∈ Ci ✱ ≤ i ≤ ✈➔ dα : S × S −→ {0; 1; 2} ❧➔ ❤➔♠ ❦❤♦↔♥❣ ❝→❝❤✳ ❚❤✉➟t t♦→♥ tr➻♥❤ ❜➔② ❝→❝❤ ❣✐↔✐ ❜➔✐ t♦→♥ õ t tổ q t ữợ ✶✿ ✣➦t S2 = S ∪ {a1 , a2 , a3 } ữợ s C1 , C2 , C3 ữợ ó S2 ✈➔ β = π − α✳ ✈➔ ✤➦t A = a1 C a2 C a3 C a1 ❧➔ ❝→❝ t➟♣ ❝♦♥ rí✐ ♥❤❛✉ ❝õ❛ A ✤✐ q✉❛ ♠å✐ ✤✐➸♠ ❝õ❛ S ♣❤➛♥ ❜ð✐ S✳ t❤➻ ❝→❝ ♣❤➛♥ ❜➔✐ t♦→♥ ♣❤➙♥ ♥❤â♠ t❛♠ ❣✐→❝✳ ◆❣÷đ❝ ❧↕✐✱ S ❧➔ ❜❛♦ ✤➣ t➼♥❤ C1 , C2 , C3 ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❦❤æ♥❣ t❤➸ ♣❤➙♥ ❤♦↕❝❤ t❤➔♥❤ ✸ dα ✳ ❉ü❛ ✈➔♦ ❜ê ✤➲ ✭✷✳✸✳✼✮ S ❤➳t t➜t ❝↔ ❝→❝ ✤✐➸♠ ❝õ❛ ♣❤➙♥ ❤♦↕❝❤ t❤➔♥❤ C1 , C2 , C3 ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ A ✤✐ q✉❛ S✳ ◆❤÷ ✈➟②✱ t❤✉➟t t♦→♥ ✷✳✸✳✽ ✤➣ ❣✐↔✐ ❜➔✐ t♦→♥ ♣❤➙♥ ♥❤â♠ t❛♠ tổ q t ữợ t ữợ tt t sỷ ❞ư♥❣ ❝→❝ t➼♥❤ t♦→♥ ❝➜♣ ✭t➼♥❤ t♦→♥ tr♦♥❣ t❤í✐ ❣✐❛♥ ✤❛ t❤ù❝✮✳ ❍ì♥ ♥ú❛✱ t➔✐ ❧✐➺✉ ❬✷❪ ✈➔ ❬✶✵❪ ✤➣ ✷✾ ❝❤➾ r❛ r➡♥❣ ❜➔✐ t♦→♥ ♣❤➙♥ ♥❤â♠ t❛♠ ❣✐→❝ ❧➔ ❧✉ỉ♥ ❣✐↔✐ ✤÷đ❝ tr♦♥❣ t❤í✐ ❣✐❛♥ ❤ú✉ ❤↕♥✳ ỵ tt tr t t♦→♥ ♣❤➙♥ ♥❤â♠ t❛♠ ❣✐→❝ ✭✤➣ ❜✐➳t✮ ✤➣ ✤÷đ❝ ✧r❡❞✉❝❡✧ ✈➲ ❜➔✐ t♦→♥ ✸ ✭❝❤÷❛ ❜✐➳t ❣✐↔✐✮✳ ❚❛ ❝â ♥❤➟♥ t s t ỗ t tt t ❣✐↔✐ ✤÷đ❝ ❜➔✐ t♦→♥ ✸ ✭t➼♥❤ ❜❛♦ ♠ët t➟♣ ❤ú✉ ❤↕♥ ✤✐➸♠ ✈ỵ✐ < α < π✮ α−❧ã♠ ❝õ❛ tr♦♥❣ t❤í✐ ❣✐❛♥ ❤ú✉ ❤↕♥✳ ❚r♦♥❣ ♠ët tr÷í♥❣ ❤đ♣ r✐➯♥❣ ❝õ❛ t➟♣ ✤✐➸♠ S✱ ❜➔✐ t♦→♥ t➼♥❤ ❜❛♦ α−❧ã♠ ❝â t❤➸ ❣✐↔✐ ✤÷đ❝ sû ❞ư♥❣ ❜➔✐ t♦→♥ ♣❤➙♥ ♥❤â♠ t❛♠ t trữợ sỷ t➟♣ ✤✐➸♠ S ❈❤♦ S ❧➔ t➟♣ ❤ú✉ ❤↕♥ ✤✐➸♠ ♥➡♠ tr♦♥❣ C1 , C2 , C3 ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ♣❤➙♥ ♥❤â♠ t❛♠ ❣✐→❝ ✈ỵ✐ ❤➔♠ dβ , β = π − α✱ ❦❤✐ ✤â t❤✉➟t t♦→♥ s❛✉ ✤➸ t➼♥❤ ♠ët ❜❛♦ α− ❧ã♠ ❝õ❛ sû ❞ö♥❣ ❜➔✐ t õ t ữợ t ởt ❜❛♦ S1 = S ∪ {a1 , a2 , a3 } ó ữợ sỷ ✈➔ β = π − α✱ ❑❤✐ ✤â t❛ ①➙② ❞ü♥❣ ❜❛♦ a1 , a2 , a3 ✈➔ ♥è✐ ❦❤✐ ✤â ✤÷í♥❣ ♥è✐ ❝→❝ ✤✐➸♠ tr♦♥❣ ✈➔ α− ❧ã♠ ❝õ❛ t➟♣ ✤✐➸♠ S x1,1 x3,r ✱ ✈ỵ✐ ♥è✐ C1 , C2 , C3 x1,p ✈ỵ✐ ❜➡♥❣ ❝→❝❤ tø ❜❛♦ x2,1 ✱ ♥è✐ ❧➔ ❜❛♦ x2,q ●✐↔✐ t❤➼❝❤ t❤✉➟t t♦→♥✳ a1 x3,r x1,1 x3,1 x2,1 ✈ỵ✐ α− ❧ã♠ A x3,1 ✳ ❚❛ ✤÷đ❝ α− ❧ã♠ ❝õ❛ t➟♣ ✤✐➸♠ S✳ x1,p ❧➛♥ ❧÷đt ❧➔✿ a3 x3,1 x3,2 x3,r a1 ✳ A = x1,1 x1,2 x1,p x2,1 x2,2 x2,q x3,1 x3,2 x3,r a2 A = a1 C a2 C a3 C S1 ✳ a1 x1,1 x1,2 x1,p a2 ✱ a2 x2,1 x2,2 x2,q a3 ❜ä ✤✐ ∆a1 a2 a3 ✳ ❈❤♦ α ❧➔ ❣â❝ x2,q a3 ✸✵ ❚r♦♥❣ ✤❛ ❣✐→❝ A = x1,1 x1,2 x1,p x2,1 x2,2 x2,q x3,1 x3,2 x3,r ✱ ❣â❝ tr♦♥❣ t↕✐ ❝→❝ ✤➾♥❤ x3,2 , x3,3 , , x3,r−1 x1,2 , x1,3 , , x1,p−1 , x2,2 , x2,3 , , x2,q−1 , ❉♦ ✤â A ❧➔ ♠ët α− AA ∆a2 x1,p x2,1 ✈➔ ✤❛ ❣✐→❝ ❝❤ù❛ ✈➔ AA A ❧➔ ▼➔ ❧➔ α− ✤❛ ❣✐→❝✮✳ ✤➲✉ ♥❤ä ❤ì♥ ❝→❝ ❣â❝ tr♦♥❣ S✳ α− ✤❛ ❣✐→❝ ❝â ❞✐➺♥ t➼❝❤ ❜➨ ♥❤➜t ❝❤ù❛ ❧➛♥ ❧÷đt ❧➔ ❞✐➺♥ t➼❝❤ ❝õ❛ ❝→❝ ✤❛ ❣✐→❝ A ✈➔ ❧➛ ❧÷đt ❧➔ ❞✐➺♥ t➼❝❤ ❝→❝ t❛♠ ❣✐→❝ S✳ A✳ ∆a1 x1,1 x3,r , ∆a3 x2,q x3,1 ✳ ❑❤✐ ✤â✱ ❞✐➺♥ t➼❝❤ ❝→❝ t❛♠ ❣✐→❝ ❤❛② ❝→❝ ❣✐→ trà A x1,1 , x1,p , x2,1 , x2,q , x3,1 , x3,r Aa1 x1,1 x3,r , Aa2 x1,p x2,1 , Aa3 x2,q x3,1 ●å✐ ✭❞♦ α− ✤❛ ❣✐→❝ A ♥➯♥ ✤➲✉ ❦❤æ♥❣ q✉→ π + α ✭❞♦ A ❧➔ α− ✤❛ ❣✐→❝✮ ❚✐➳♣ t❤❡♦ t❛ ❝❤ù♥❣ ♠✐♥❤ ❚❤➟t ✈➟②✿ ●å✐ π+α ✤➲✉ ❦❤æ♥❣ q✉→ ❈→❝ ❣â❝ tr♦♥❣ t↕✐ ❝→❝ ✤➾♥❤ t↕✐ ❝→❝ ✤➾♥❤ ✤â ❝õ❛ ❤✐➸♥ ♥❤✐➯♥ ❝→❝ ∆a1 x1,1 x3,r , ∆a2 x1,p x2,1 Aa1 x1,1 x3,r , Aa2 x1,p x2,1 , Aa3 x2,q x3,1 ✈➔ ∆a3 x2,q x3,1 ❝è ✤à♥❤✱ ❦❤æ♥❣ ✤ê✐✳ AA = AA − Aa1 x1,1 x3,r + Aa2 x1,p x2,1 + Aa3 x2,q x3,1 ◆➯♥ AA ❜❛♦ α− ❉♦ ✤â ♥❤ä ♥❤➜t ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ❧ã♠ ❝õ❛ t➟♣ ✤✐➸♠ A ❧➔ α− AA ♥❤ä ♥❤➜t✳ ✣✐➲✉ ♥➔② ❤✐➸♥ ♥❤✐➯♥ ✤ó♥❣ ✈➻ A ❧➔ S ∪ {a1 , a2 , a3 }✳ ✤❛ ❣✐→❝ ❝â ❞✐➺♥ t➼❝❤ ❜➨ ♥❤➜t ❝❤ù❛ t➟♣ ✤✐➸♠ ❍❛② A = x1,1 x1,2 x1,p x2,1 x2,2 x2,q x3,1 x3,2 x3,r ✤✐➸♠ S✳ S✳ ❧➔ ❜❛♦ α− ❧ã♠ ❝õ❛ t➟♣ ❑➳t ❧✉➟♥ ✣➲ t➔✐ ✤➣ t❤✉ ✤÷đ❝ ❝→❝ ❦➳t q✉↔ s❛✉ ✤➙②✿ ✲ ❍➺ t❤è♥❣ ❝→❝ ❦❤→✐ ♥✐➺♠✱ ❝→❝ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ✈➔ ❝→❝ ✈➼ ❞ư ♠✐♥❤ ❤å❛ ✈➲✿ t➟♣ ỗ ỗ ỡ ã ◆➯✉ r❛ ❝→❝ ♥❤➟♥ ①➨t ✈➲ ❜❛♦ • ◆➯✉ r❛ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❜❛♦ α✲❧ã♠ α✲❧ã♠✳ ✣➦❝ ❜✐➺t✿ ✭◆❤➟♥ ①➨t ✷✳✶✳✷✱ ✷✳✶✳✸✮✳ α✲❧ã♠ ✭❚➼♥❤ ❝❤➜t ✷✳✷✳✶ ✲ ✷✳✷✳✸✮ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t ❚➼♥❤ ❝❤➜t ✷✳✷✳✸ ♠➔ t➔✐ ❧✐➺✉ ❬✶❪ ❝❤➾ ❝❤ù♥❣ ♠✐♥❤ ❧÷đ❝✳ ✲ ❚r➻♥❤ ❜➔② ❚❤✉➟t t♦→♥ ✷✳✸✳✶ ❝❤✉②➸♥ ✤ê✐ ❜➔✐ t♦→♥ tê♥❣ t➟♣ ❝♦♥ ✤➣ ❜✐➳t ✈➲ t➼♥❤ α✲✤❛ ❣✐→❝ ❝õ❛ ♠ët t➟♣ ✤✐➸♠ ợ t trữợ t t ✤ê✐ ❜➔✐ t♦→♥ ♣❤➙♥ ♥❤â♠ t❛♠ ❣✐→❝ ✈➲ t➼♥❤ ❜❛♦ ợ ã < < ó ❝õ❛ ♠ët t➟♣ ✤✐➸♠ tr♦♥❣ ♠➦t ✣➦❝ ❜✐➺t✿ ❚r➻♥❤ ❜➔② ❝❤✐ t✐➳t ❝❤ù♥❣ ♠✐♥❤ ❝õ❛ ❚❤✉➟t t♦→♥ ✷✳✸✳✶ ✈➔ ❱➼ ❞ư ✷✳✸✳✷ ♠➔ t➔✐ ❧✐➺✉ ❬✶❪ ❦❤ỉ♥❣ ❝❤ù♥❣ ♠✐♥❤✳ ❚r➻♥❤ ❜➔② ❝❤✐ t✐➳t ❝❤ù♥❣ ♠✐♥❤ ❇ê ✤➲ ✷✳✸✳✼ ♠➔ t➔✐ ự sỡ ữủ ã r ❝❤✐ t✐➳t ❚❤✉➟t t♦→♥ ✷✳✸✳✽ ♠➔ t➔✐ ❧✐➺✉ ❬✶❪ ❝❤➾ ✤➲ ❝➟♣ ♥❣➢♥ ❣å♥✳ ✸✶ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ❙✳ ❆s❛❡❡❞✐✱ ❋✳ ❉✐❞❡❤✈❛r✱ ❆✳ ▼♦❤❛❞❡s ✭✷✵✶✼✮✱ α✲❈♦♥❝❛✈❡ ❤✉❧❧✱ ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ❝♦♥✈❡① ❤✉❧❧✱ ❚❤❡♦r❡t✐❝❛❧ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ✼✵✷✱ ✹✽✲✺✾✳ ❬✷❪ P✳ ❇r✉❝❦❡r ✭✶✾✼✽✮✱ ❖♥ t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ ❝❧✉st❡r✐♥❣ ♣r♦❜❧❡♠s✱ ✐♥✿ ❖♣t✐♠✐③❛t✐♦♥ ❛♥❞ ❖♣❡r❛t✐♦♥s ❘❡s❡❛r❝❤✱ ❙♣r✐♥❣❡r ❬✸❪ ❉✳❘✳ ❈❤❛♥❞✱ ❙✳❙✳ ❑❛♣✉r ✭✶✾✼✵✮✱ ❆♥ ❛❧❣♦r✐t❤♠ ❢♦r ❝♦♥✈❡① ♣♦❧②t♦♣❡s✱ ❏✳ ❆❈▼✱ ✼✽✲✽✻✳ ❬✹❪ ❆✳▲✳ ❈❤❛✉✱ ❳✳ ▲✐✱ ❲✳ ❨✉ ✭✷✵✶✸✮✱ ▲❛r❣❡ ❞❛t❛ s❡ts ❝❧❛ss✐❢✐❝❛t✐♦♥ ✉s✐♥❣ ❝♦♥✲ ✈❡①✕❝♦♥❝❛✈❡ ❤✉❧❧ ❛♥❞ s✉♣♣♦rt ✈❡❝t♦r ♠❛❝❤✐♥❡✱ ❙♦❢t✳ ❈♦♠♣✉t✳✱ ✼✾✸✲✽✵✹✳ ❬✺❪ ❆✳ ●❛❧t♦♥✱ ▼✳ ❉✉❝❦❤❛♠ ✭✷✵✵✻✮✱ ❲❤❛t ✐s t❤❡ r❡❣✐♦♥ ♦❝❝✉♣✐❡❞ ❜② ❛ s❡t ♦❢ ♣♦✐♥ts✱ ●❡♦❣r❛♣❤✐❝ ■♥❢♦r♠❛t✐♦♥ ❙❝✐❡♥❝❡✱ ❙♣r✐♥❣❡r✱ ✽✶✲✾✽✳ ❬✻❪ ❘✳▲✳ ●r❛❤❛♠ ✭✶✾✼✷✮✱ ❆♥ ❡❢❢✐❝✐❡♥t ❛❧❣♦r✐t❤♠ ❢♦r ❞❡t❡r♠✐♥✐♥❣ t❤❡ ❝♦♥✈❡① ❤✉❧❧ ♦❢ ❛ ❢✐♥✐t❡ ♣❧❛♥❛r s❡t✱ ■♥❢♦r♠✳ Pr♦❝❡ss✳ ▲❡tt✱ ✶✸✷✲✶✸✸✳ ❬✼❪ ❏✳ ❏♦♥❡s ✭✷✵✶✻✮✱ ▼✉❧t✐✲❛❣❡♥t s❧✐♠❡ ♠♦✉❧❞ ❝♦♠♣✉t✐♥❣✿ ♠❡❝❤❛♥✐s♠s✱ ❛♣♣❧✐❝❛✲ t✐♦♥s ❛♥❞ ❛❞✈❛♥❝❡s✱ ❆❞✈❛♥❝❡s ✐♥ P❤②s❛r✉♠ ▼❛❝❤✐♥❡s✱ ❙♣r✐♥❣❡r✳ ❬✽❪ ■✳ ❊✳ ▲❡♦♥❛r❞✱ ❏✳ ❊✳ ▲❡✇✐s ✭✷✵✶✺✮✱ ●❡♦♠❡tr② ♦❢ ❈♦♥✈❡① ❙❡ts ✱ ❏♦❤♥ ❲✐❧❡② ❛♥❞ ❙♦♥s✱ ✶✸✺✲✶✻✾✳ ❬✾❪ ❆✳ ▼♦r❡✐r❛✱ ▼✳❨✳ ❙❛♥t♦s ✭✷✵✵✼✮✱ ❈♦♥❝❛✈❡ ❤✉❧❧✿ ❆ ❦✲♥❡❛r❡st ♥❡✐❣❤❜♦✉rs ❛♣✲ ♣r♦❛❝❤ ❢♦r t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ r❡❣✐♦♥ ♦❝❝✉♣✐❡❞ ❜② ❛ s❡t ♦❢ ♣♦✐♥ts✱ ■◆❙❚■❈❈ Pr❡ss✳ ✸✷ ✸✸ ❬✶✵❪ ❘✳●✳ ▼✐❝❤❛❡❧✱ ❙✳❏✳ ❉❛✈✐❞✱ ❈♦♠♣✉t❡rs ❛♥❞ ■♥tr❛❝t❛❜✐❧✐t②✿ ❆ ●✉✐❞❡ t♦ t❤❡ ❚❤❡✲ ♦r② ♦❢ ◆P✲❈♦♠♣❧❡t❡♥❡ss✱ ❲✳❍✳ ❋r❡❡♠❛♥ ❈♦✱ ❙❛♥ ❋r❛♥❝✐s❝♦✳ ❬✶✶❪ ❏✳✲❙✳ P❛r❦✱ ❙✳✲❏✳ ❖❤ ✭✷✵✶✸✮✱ ❆ ♥❡✇ ❝♦♥❝❛✈❡ ❤✉❧❧ ❛❧❣♦r✐t❤♠ ❛♥❞ ❝♦♥❝❛✈❡♥❡ss ♠❡❛s✉r❡ ❢♦r ♥✲❞✐♠❡♥s✐♦♥❛❧ ❞❛t❛s❡ts✱ ❏■❙❊ ❏✳ ■♥❢✳ ❙❝✐✳ ❊♥❣✱ ✸✼✾✲✸✾✷✳ ❬✶✷❪ ❋✳P✳ Pr❡♣❛r❛t❛✱ ❙✳❏✳ ❍♦♥❣ ✭✶✾✼✼✮✱ ❈♦♥✈❡① ❤✉❧❧s ♦❢ ❢✐♥✐t❡ s❡ts ♦❢ ♣♦✐♥ts ✐♥ t✇♦ ❛♥❞ t❤r❡❡ ❞✐♠❡♥s✐♦♥s✱ ❈♦♠♠✉♥✳ ❆❈▼✱ ✽✼✲✾✸✳ ❬✶✸❪ ❏✳ ❖✬❘♦✉r❦❡ ✭✶✾✾✽✮✱ ❈♦♠♣✉t❛t✐♦♥❛❧ ●❡♦♠❡tr② ✐♥ ❈✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✳ ❬✶✹❪ ❉✳◆✳ ❙✐r✐❜❛✱ ❙✳▼✳ ▼❛t❛r❛✱ ❙✳▼✳ ▼✉s②♦❦❛ ✭✷✵✶✺✮✱ ■♠♣r♦✈❡♠❡♥t ♦❢ ✈♦❧✉♠❡ ❡st✐✲ ♠❛t✐♦♥ ♦❢ st♦❝❦♣✐❧❡ ♦❢ ❡❛rt❤✇♦r❦s ✉s✐♥❣ ❛ ❝♦♥❝❛✈❡ ❤✉❧❧✲❢♦♦t♣r✐♥t✱ ■♥t❡r♥❛t✐♦♥❛❧ ❙❝✐✳❏✳ ▼✐❝r♦ ▼❛❝r♦ ▼❡③③♦ ●❡♦✐♥❢✱ ✶✶✲✷✺✳ ❬✶✺❪ ●✳❚✳ ❚♦✉ss❛✐♥t ✭✶✾✽✺✮✱ ❆ ❤✐st♦r✐❝❛❧ ♥♦t❡ ♦♥ ❝♦♥✈❡① ❤✉❧❧ ❢✐♥❞✐♥❣ ❛❧❣♦r✐t❤♠s✱ P❛tt❡r♥ ❘❡❝♦❣♥✐t✳ ▲❡tt✱ ✷✶✲✷✽✳ ❬✶✻❪ ❆✳ ❱✐s❤✇❛♥❛t❤✱ ▼✳ ❘❛♠❛♥❛t❤❛♥ ✭✷✵✶✷✮✱ ❈♦♥❝❛✈❡ ❤✉❧❧ ♦❢ ❛ s❡t ♦❢ ❢r❡❡❢♦r♠ ❝❧♦s❡❞ s✉r❢❛❝❡s ✐♥ R3 ✱ ❈♦♠♣✉t✲❆✐❞❡❞ ❉❡s✳ ❆♣♣❧✱ ✽✺✼✲✽✻✽✳