Carath´ eodory’s theorem(ΧϥςΦυϦͷఆཧ) ҙͷू߹ S ⫅ Rn ͷತแ co S ɼS ʹଐ͢Δߴʑ n + 1 ݸͷͷ ತ݁߹શମͷू߹ *2) ʹ͍͠ɽ • ิ 2.2 ΑΓɼ༗ݶݸͷͷತ݁߹ɼͦΕΒͷͷதͷߴʑ n + 1 ݸͷͷತ݁߹Ͱද͢͜ͱ͕Ͱ͖Δɽ • ͕ͨͬͯ͠ɼຊఆཧΛࣔͨ͢Ίʹɼू߹ S ʹଐ͢Δ༗ݶݸͷ ͷತ݁߹શମͷू߹͕ co S ʹҰக͢Δ͜ͱΛࣔͤΑ͍ɽ *2)͜Ε S ʹଐ͢Δߴʑ n + 1 ݸͷͷબͼํͱͦΕΒͷͷತ݁߹ʹ͓͚Δͷબͼ ํͷ྆ํʹؔͯ͠ɼͯ͢ͷՄೳੑΛߟ͑Δ͜ͱΛҙຯ͍ͯ͠Δɽ 13 / 16
2.2 ʹର͢Δূ໌Λ༩͑Δɽ࣍ͷ 2 ͭͷ part ʹ͚Δɿ 1. S ͷ༗ݶݸͷͷತ݁߹શମͷू߹͕ co S ʹؚ·ΕΔ͜ͱ 2. S ͷ༗ݶݸͷͷತ݁߹શମͷू߹͕ತͰ͋Δ͜ͱ ূ໌ɽ[part1] S ʹଐ͢Δҙͷ x1, . . . , xm ʹରͯ͠ɼ x = ∑ m i=1 αixi ∈ co S Ͱ͋Δ͜ͱΛɼm ʹؔ͢Δֶతؼೲ๏Ͱ ࣔ͢ɽm = 1 ͷͱ͖ɼx = x1 ∈ S ⫅ co S Ͱ͋Δɽ࣍ʹɼS ͷҙ ͷ m ݸͷͷತ݁߹͕ co S ʹଐ͢Δ (˒) ͱԾఆͯ͠ɼҙͷ m + 1 ݸͷ x1, . . . , xm+1 ∈ S ͱ ∑ m+1 i=1 αi = 1 Λຬͨ͢ҙͷ ඇෛ࣮ αi ≧ 0 (i = 1, . . . , m + 1) ʹରͯ͠ɼ x = ∑ m+1 i=1 αixi ͕ co S ʹଐ͢Δ͜ͱΛࣔ͢ɽ 14 / 16