ඇઢܗ࠷దԽͷجૅ – Carath´ eodory’s theorem – miruca Graduate School of Informatics, Kyoto University March 13, 2019

͜ͷεϥΠυͷ໨త ʰඇઢܗ࠷దԽͷجૅʱ(෱ౡ, 2001) ʹؔͯ͠ • ತղੳͷجຊతͳ಺༰ (e.g., ತू߹ɼತแɼತ݁߹) Λཧղ͢Δ • ತ݁߹ʹؔ͢Δ༗༻ͳิ୊ (ิ୊ 2.2) Λཧղ͢Δ • Caratheodory’s theorem(ΧϥςΦυϦͷఆཧ) Λཧղ͢Δ ˞஫ҙ • ຊεϥΠυͷఆཧ౳ͷ൪߸͸ʰඇઢܗ࠷దԽͷجૅʱʹ४ͣΔ • ਤ͸ͳ͍ͷͰదٓखΛಈ͔͠ͳ͕Βཧղ͢Δ͜ͱΛਪ঑ • ΧϥςΦυϦͷఆཧʹؔͯ͠ɼࣗ෼ͰྫΛߏ੒ͯ͠ཧղΛਂΊ Δ͜ͱΛਪ঑

ತू߹ Carath´ eodory’s theorem Today’s Topic 1. ತू߹ 2. Carath´ eodory’s theorem 3 / 16

1. ತू߹ 2. Carath´ eodory’s theorem

ತू߹ Carath´ eodory’s theorem ತू߹ ఆٛ: ತू߹ (convex set) ू߹ S ⫅ Rn ͷ೚ҙͷ 2 ఺Λ݁Ϳઢ෼͕ू߹ S ʹؚ·ΕΔɼ͢ͳΘ ͪɼ೚ҙͷ࣮਺ α ∈ [0, 1] ʹରͯ͠ x ∈ S, y ∈ S ⇒ (1 − α)x + αy ∈ S ͕੒ΓཱͭͳΒ͹ɼS ͸ತू߹ (convex set) Ͱ͋Δͱ͍͏ɽ ྫ: ತू߹ • C1 = { (x1, x2) ∈ R2 | x2 1 + x2 2 ≦ 1 } • C2 = { x ∈ Rn | x = ∑ m i=1 αiai, αi ≧ 0 (i = 1, . . . , m) } • C3 = { (x1, x2, x3) ∈ R3 | x2 1 ≧ x2 2 + x2 3 , x1 ≧ 0 } 5 / 16

ತू߹ Carath´ eodory’s theorem ತू߹ʹؔ͢Δఆཧ ఆཧ 2.1 ೚ҙݸͷತू߹ Si (i ∈ I) ͷڞ௨ू߹ ∩i∈ISi ͸ತू߹Ͱ͋Δɽ ͜͜ͰɼI ͸೚ҙͷఴࣈू߹ *1) Ͱ͋Δɽ ূ໌. ೚ҙݸͷತू߹ Si (i ∈ I) ͷڞ௨ू߹ ∩i∈ISi Λ S Ͱද͢ɽ ͭ·ΓɼS = ∩i∈ISi ͱ͢ΔɽS ʹଐ͢Δ 2 ఺ x, y ͸ɼ͢΂ͯͷ i ∈ I ʹରͯ͠ɼx, y ∈ Si Ͱ͋Γɼ֤ Si ͸ԾఆΑΓತू߹Ͱ͋Δ ͔Βɼ೚ҙͷ α ∈ [0, 1] ʹରͯ͠ɼ(1 − α)x + αy ∈ Si ͕੒Γཱͭɽ Αͬͯɼ೚ҙͷ α ∈ [0, 1] ʹରͯ͠ɼ(1 − α)x + αy ∈ ∩i∈ISi = S ͱͳΔͷͰɼS ͸ತू߹Ͱ͋Δɽ(ূ໌ऴ) ˞͢΂ͯͷू߹ Si (i ∈ I) ͕ดू߹Ͱ͋Ε͹ɼू߹ S ΋ดू߹ɽ *1)ఴࣈू߹ͱ͸ɼྫ͑͹ɼ༗ݶݸͷཁૉΛؚΉ I = {1, 2, . . . , m} Ͱ͋ͬͨΓɼແݶݸͷ ཁૉΛؚΉ N(ࣗવ਺શମͷू߹) ͳͲΛද͢ɽ 6 / 16

ತू߹ Carath´ eodory’s theorem ತแɾತ݁߹ ఆٛ: ತแ (convex hull) ೚ҙͷू߹ S ⫅ Rn ʹରͯ͠ɼS ΛؚΉ࠷খͷತू߹Λ S ͷತแ (convex hull) ͱݺͼɼco S Ͱද͢ɽ ఆٛ: ತ݁߹ (convex combination) m ݸͷ఺ x1, . . . , xm ∈ Rn ʹରͯ͠ɼ α1 + · · · + αm = 1 Λຬͨ͢Α͏ͳඇෛ࣮਺ αi ≧ 0 (i = 1, . . . , m) Λ༻͍ͯ x = α1x1 + · · · + αmxm ͱද͞ΕΔϕΫτϧ x ∈ Rn Λ x1, . . . , xm ∈ Rn ͷತ݁߹ (convex combination) ͱ͍͏ɽ 7 / 16

ತू߹ Carath´ eodory’s theorem ತ݁߹ʹؔ͢Δิ୊ ิ୊ 2.2 ఺ x ∈ Rn ͕ m ݸͷ఺ x1, . . . , xm ∈ Rn ͷತ݁߹ͱͯ͠ද͞Εͯ ͍Δͱ͢Δɽ͜ͷͱ͖ɼm ≧ n + 2 ͳΒ͹ɼ఺ x1, . . . , xm ∈ Rn ͔ Βߴʑ n + 1 ݸͷ఺ΛબΜͰɼx ΛͦΕΒͷತ݁߹ͱͯ͠ද͢͜ͱ ͕Ͱ͖Δɽ ূ໌. {x1, . . . , xm} ͷ෦෼ू߹Ͱɼx Λತ݁߹ͱͯ͠ද͢͜ͱͷͰ ͖Δ΋ͷͷ͏ͪɼͦͷϕΫτϧͷݸ਺͕࠷খͷ΋ͷΛɼҰൠੑΛࣦ ͏͜ͱͳ͘ɼ{x1, . . . , xp} ͱͰ͖Δɽͨͩ͠ɼp ≦ m Ͱ͋Δɽ͍ ·ɼิ୊ 2.2 ͕ਖ਼͘͠ͳ͍ͱԾఆ͢Δͱɼp ≧ n + 2 Ͱ͋ͬͯ x = p ∑ i=1 αixi ͔ͭ ∑ p i=1 αi = 1 ͳΔ αi > 0 (i = 1, . . . , p) ͕ଘࡏ͢Δ (˞ 1)ɽ 8 / 16

ತू߹ Carath´ eodory’s theorem ͜͜ͰɼϕΫτϧ xi − xp (i = 1, . . . , p − 1) Λߟ͑Δͱɼp ≧ n + 2 ΑΓɼp − 1 ≧ n + 1 ≧ n Ͱ͋Δ͔Βɼ͜ΕΒͷϕΫτϧ͸Ұ࣍ैଐ Ͱ͋Δ (˞ 2)ɽΑͬͯɼগͳ͘ͱ΋Ұͭ͸ਖ਼Ͱ͋ΔΑ͏ͳ࣮਺ β1, . . . , βp−1 ʹରͯ͠ɼ p−1 ∑ i=1 βi(xi − xp) = p−1 ∑ i=1 βixi + ( − p−1 ∑ i=1 βi ) xp = 0 ͕੒Γཱͭɽ͜͜Ͱɼβp := − ∑ p−1 i=1 βi ͱ͓͘ͱ p−1 ∑ i=1 βixi + ( − p−1 ∑ i=1 βi ) xp = p−1 ∑ i=1 βixi + βpxp = p ∑ i=1 βixi = 0 ͕੒Γཱͭɽ 9 / 16

ತू߹ Carath´ eodory’s theorem ·ͨɼβp = − ∑ p−1 i=1 βi ΑΓɼ ∑ p i=1 βi = 0 Ͱ͋Δɽ͕ͨͬͯ͠ɼ೚ ҙͷ࣮਺ τ ʹରͯ͠ɼx ∈ Rn ͸࣍ͷΑ͏ʹදͤΔɽ x = p ∑ i=1 (αi − τβi) xi ͜͜Ͱɼτ := min{αi/βi | βi > 0} ͱ͠ɼα′ i := αi − τβi ͱ͓͘ɽ ͢Δͱɼx = ∑ p i=1 α′ i xi, ∑ p i=1 α′ i = 1 α′ i ≧ 0 ͕͔֬ΊΒΕΔ (˞ 3)ɽ͞Βʹɼτ ͷఆΊํΑΓɼগͳ͘ͱ΋Ұͭͷ j (1 ≦ j ≦ m) ʹ ରͯ͠ɼα′ j = 0 Ͱ͋Δ (˞ 4)ɽ͜Ε͸ɼx ͕࣮࣭తʹ p − 1 ݸͷ఺ ͷತ݁߹ͰදͤΔ͜ͱΛҙຯ͍ͯ͠Δ (˞ 5)ɽx Λತ݁߹Ͱදͨ͢ ΊʹඞཁͳϕΫτϧͷ࠷খݸ਺Λ p ݸͱԾఆͨ͠ͷͰɼ͜Ε͸ໃ६ Ͱ͋Δɽ͕ͨͬͯ͠ɼx ͸ߴʑ n + 1 ݸͷ఺ xi ͷತ݁߹ͱͯ͠ද ͤΔɽ(ূ໌ऴ) 10 / 16

ತू߹ Carath´ eodory’s theorem ิ଍ ˞ 1. ิ୊Ͱ͸ߴʑ n + 1 ݸͷ఺Ͱे෼ͱओு͍ͯ͠Δ͕ɼূ໌Ͱ͸ p(≧ n + 2) ݸͷ఺͕ඞཁͱԾఆͯ͠ໃ६Λಋ͜͏ͱ͍ͯ͠Δɽ ·ͨɼαi > 0 ͱͯ͠΋Α͍ͷ͸ɼαi = 0 ͳΔ αi ΛऔΓআ͍ͯ ΋ x ͸ͦΕҎ֎ͷ఺ͷತ݁߹ͱͯ͠ද͢͜ͱ͕Ͱ͖ΔͨΊɽ ˞ 2. x ∈ Rn ͳͷͰɼx ͸ߴʑ n ݸͷϕΫτϧͷઢܗ࿨Ͱද͢͜ͱ ͕Ͱ͖Δɽ ˞ 3. ∑ p i=1 βi = 0 Ͱ͋Δ͜ͱʹ஫ҙ͢Δͱ͍ͣΕ΋༰қʹ͔֬ΊΒ ΕΔɽ ˞ 4. τ = αj Ͱ͋Δ (i = j ͷͱ͖ʹ αi/βi ͕࠷খͱͳΔ) ͱ͢Δͱɼ α′ j = αj − αj/βj · βj = 0 ͱͳΔɽ ˞ 5. α′ j = 0 ͳͷͰɼα′ j ΛऔΓআ͍ͯ΋ x ͸ͦΕҎ֎ͷ p − 1 ݸͷ ఺ͷತ݁߹ͱͯ͠ද͢͜ͱ͕Ͱ͖Δɽ 11 / 16

1. ತू߹ 2. Carath´ eodory’s theorem

ತू߹ Carath´ eodory’s theorem Carath´ eodory’s theorem ΧϥςΦυϦͷఆཧͷεςʔτϝϯτ͸ҎԼͰ͋Δɽ ఆཧ 2.2: Carath´ eodory’s theorem(ΧϥςΦυϦͷఆཧ) ೚ҙͷू߹ S ⫅ Rn ͷತแ co S ͸ɼS ʹଐ͢Δߴʑ n + 1 ݸͷ఺ͷ ತ݁߹શମͷू߹ *2) ʹ౳͍͠ɽ • ิ୊ 2.2 ΑΓɼ༗ݶݸͷ఺ͷತ݁߹͸ɼͦΕΒͷ఺ͷதͷߴʑ n + 1 ݸͷ఺ͷತ݁߹Ͱද͢͜ͱ͕Ͱ͖Δɽ • ͕ͨͬͯ͠ɼຊఆཧΛࣔͨ͢Ίʹ͸ɼू߹ S ʹଐ͢Δ༗ݶݸͷ ఺ͷತ݁߹શମͷू߹͕ co S ʹҰக͢Δ͜ͱΛࣔͤ͹Α͍ɽ *2)͜Ε͸ S ʹଐ͢Δߴʑ n + 1 ݸͷ఺ͷબͼํͱͦΕΒͷ఺ͷತ݁߹ʹ͓͚Δ܎਺ͷબͼ ํͷ྆ํʹؔͯ͠ɼ͢΂ͯͷՄೳੑΛߟ͑Δ͜ͱΛҙຯ͍ͯ͠Δɽ 13 / 16

ತू߹ Carath´ eodory’s theorem Proof of Carath´ eodory’s theorem ఆཧ 2.2 ʹର͢Δূ໌Λ༩͑Δɽ࣍ͷ 2 ͭͷ part ʹ෼͚Δɿ 1. S ͷ༗ݶݸͷ఺ͷತ݁߹શମͷू߹͕ co S ʹؚ·ΕΔ͜ͱ 2. S ͷ༗ݶݸͷ఺ͷತ݁߹શମͷू߹͕ತͰ͋Δ͜ͱ ূ໌ɽ[part1] S ʹଐ͢Δ೚ҙͷ఺ x1, . . . , xm ʹରͯ͠ɼ x = ∑ m i=1 αixi ∈ co S Ͱ͋Δ͜ͱΛɼm ʹؔ͢Δ਺ֶతؼೲ๏Ͱ ࣔ͢ɽm = 1 ͷͱ͖ɼx = x1 ∈ S ⫅ co S Ͱ͋Δɽ࣍ʹɼS ͷ೚ҙ ͷ m ݸͷ఺ͷತ݁߹͕ co S ʹଐ͢Δ (˒) ͱԾఆͯ͠ɼ೚ҙͷ m + 1 ݸͷ఺ x1, . . . , xm+1 ∈ S ͱ ∑ m+1 i=1 αi = 1 Λຬͨ͢೚ҙͷ ඇෛ࣮਺ αi ≧ 0 (i = 1, . . . , m + 1) ʹରͯ͠ɼ఺ x = ∑ m+1 i=1 αixi ͕ co S ʹଐ͢Δ͜ͱΛࣔ͢ɽ 14 / 16

ತू߹ Carath´ eodory’s theorem αm+1 = 1 ͷ৔߹͸໌Β͔Ͱ͋ΔͷͰɼαm+1 < 1 ͷ৔߹ʹ͍ͭͯ ߟ͑Δɽ͜ͷͱ͖ɼ఺ x ͸࣍ͷΑ͏ʹද͢͜ͱ͕Ͱ͖Δɽ x = (1 − αm+1) m ∑ i=1 αi 1 − αm+1 xi + αm+1xm+1 ͜͜Ͱɼβi = αi/(1 − αm+1) ͱ͓͘ͱ m ∑ i=1 βi = 1 1 − αm+1 m ∑ i=1 αi = 1 − αm+1 1 − αm+1 = 1 βi ≧ 0 (i = 1, . . . , m) ͕͔֬ΊΒΕΔɽؼೲ๏ͷԾఆ (˒) ΑΓɼ͕࣍ࣜ੒Γཱͭɿ ˜ x := m ∑ i=1 βixi = 1 1 − αm+1 m ∑ i=1 αixi ∈ co S. ·ͨɼxm+1 ∈ co S ͔ͭ co S ͸ತू߹Ͱ͋Δ͔Βɼ (1 − αm+1)˜ x + αm+1xm+1 ∈ co S Ͱ͋Δ (ತू߹ͷఆٛ)ɽ 15 / 16

ತू߹ Carath´ eodory’s theorem ࣍ʹ S ͷ༗ݶݸͷ఺ͷತ݁߹શମͷू߹͕ತͰ͋Δ͜ͱΛࣔ͢ *3)ɽ S ͷ༗ݶݸͷ఺ͷತ݁߹શମͷू߹Λ S′ ͱද͠ɼS′ ʹؚ·ΕΔ೚ ҙͷ 2 ఺ x, y ΛબͿɽ͜ͷͱ͖ɼ఺ x ͱ y ͸ͦΕͧΕ S ͷ఺ xi (i = 1, . . . , m) ͱ఺ yj (j = 1, . . . , l) ͷತ݁߹ͱͯ͠ɼ x = ∑ m i=1 αixi, y = ∑ l j=1 αjxj ͷΑ͏ʹද͞ΕΔɽ͞Βʹɼ೚ҙ ͷ࣮਺ λ ∈ [0, 1] ʹରͯ͠ɼ఺ z = (1 − λ)x + λy Λߟ͑Δͱɼ࣍ ͕ࣜ੒Γཱͭɿ m ∑ i=1 (1 − λ)αi + l ∑ j=1 λβj = (1 − λ) m ∑ i=1 αi 1 +λ l ∑ j=1 βj 1 = 1. S′ ͷఆٛΑΓ z ∈ S′ Ͱ͋Γɼू߹ S′ ͸ತू߹Ͱ͋Δɽ(ূ໌ऴ) *3)ತแͷఆٛΑΓ co S ͸ “S ΛؚΉ࠷খͷತू߹” Ͱ͋ΔɽΑͬͯɼS ͷ༗ݶݸͷ఺ͷತ ݁߹શମͷू߹͕ತͰ͋Δ͜ͱ͕ݴ͑Ε͹ɼͦͷू߹͕ co S ͱҰக͢Δ͜ͱ͕ؼ݁Ͱ͖Δɽ 16 / 16