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❖♣t✐♠❛❧ ❈♦✈❡r✐♥❣ ♦❢ ❛ ❉✐s❦ ✇✐t❤ ❈♦♥❣r✉❡♥t ❙♠❛❧❧❡r ❉✐s❦s ❇❛❧á③s ❈s✐❦ós ❊öt✈ös ▲♦rá♥❞ ❯♥✐✈❡rs✐t② ❇✉❞❛♣❡st ❘❡tr♦s♣❡❝t✐✈❡ ❲♦r❦s❤♦♣ ♦♥ ❉✐s❝r❡t❡ ●❡♦♠❡tr②✱ ❖♣t✐♠✐③❛t✐♦♥✱ ❛♥❞ ❙②♠♠❡tr② ◆♦✈❡♠❜❡r ✷✺✲✷✾✱ ✷✵✶✸ ❋✐❡❧❞s ■♥st✐t✉t❡✱ ❚♦r♦♥t♦

Pr♦❜❧❡♠✿ ❋✐♥❞ r n p = max { r : n ❞✐s❦s ♦❢ r❛❞✐✉s r ❝❛♥ ❜❡ ♣❛❝❦❡❞ ✐♥t♦ ❛ ✉♥✐t ❞✐s❦✳ } ✭✶✲✹ tr✐✈✐❛❧❀ ✺✲✼ ●r❛❤❛♠ ✭✶✾✻✽✮❀ ✽✲✶✵ P✐r❧ ✭✶✾✻✾✮❀ ✶✶ ▼❡❧✐ss❡♥ ✭✶✾✾✹✮❀ ✶✷ ❋♦❞♦r ✭✷✵✵✵✮❀ ✶✸ ❋♦❞♦r ✭✷✵✵✸✮❀ ✶✹ ✉♥s♦❧✈❡❞✮

❈♦✈❡r✐♥❣ ♦❢ ❛ ❉✐s❦ ✇✐t❤ ❈♦♥❣r✉❡♥t ❉✐s❦s Pr♦❜❧❡♠✿ ❋✐♥❞ r n c = min { r : n ❞✐s❦s ♦❢ r❛❞✐✉s r ❝❛♥ ❝♦✈❡r ❛ ✉♥✐t ❞✐s❦✳ } ❙♦❧✉t✐♦♥s ❢♦r n ≤ 8 ✿ ❋♦r n = 5 ❛♥❞ 6 t❤❡ ♦♣t✐♠❛❧ ❝♦♥✜❣✉r❛t✐♦♥ ❤❛s ♦♥❧② ❛ ♠✐rr♦r s②♠♠❡tr②✳ ✭✶✲✹✱ ✼ tr✐✈✐❛❧❀ ✺✲✻ ❑✳ ❇❡③❞❡❦ ✭✶✾✼✾✱✶✾✽✸✮❀ ✽✲✾ ●✳ ❋❡❥❡s ❚ót❤ ✭✶✾✾✾✮❀ ✶✵ ❄✮

❘✳ ❈♦♥♥❡❧❧②✬s Pr♦❜❧❡♠✿ ◮ ●✐✈❡♥ n ❛♥❞ r n p ≤ r ≤ r n c ✱ ✜♥❞ t❤❡ ❝♦♥✜❣✉r❛t✐♦♥ ♦❢ n ❞✐s❦s ♦❢ r❛❞✐✉s r ✱ t❤❛t ❝♦✈❡rs t❤❡ ♠♦st ♦❢ t❤❡ ❛r❡ ♦❢ ❛ ✉♥✐t ❞✐s❦✳ ◮ ❯♥❞❡rst❛♥❞ ❤♦✇ t❤❡ r♦t❛t✐♦♥❛❧ s②♠♠❡tr② ♦❢ t❤❡ ♦♣t✐♠❛❧ ❝♦♥✜❣✉r❛t✐♦♥ ❢♦r n = 5 ✱ r = r 5 p ✐s ❧♦st ❛s r ❣r♦✇s ❝♦♥t✐♥✉♦✉s❧② ❢r♦♠ r 5 p t♦ r 5 c ✳ ??? ??? − → − →

❉❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❱♦❧✉♠❡ ♦❢ ❋❧♦✇❡rs ❉❡✜♥✐t✐♦♥s ◮ ❆ ❧❛tt✐❝❡ ♣♦❧②♥♦♠✐❛❧ f ( x 1 , . . . , x k ) ✐s ❛ ❢♦r♠❛❧ ❡①♣r❡ss✐♦♥ ❜✉✐❧t ❢r♦♠ t❤❡ ✈❛r✐❛❜❧❡s x 1 , . . . , x n ❛♥❞ t❤❡ ❜✐♥❛r② ♦♣❡r❛t✐♦♥ s②♠❜♦❧s ∧ ❛♥❞ ∨ ✳ ❊①❛♠♣❧❡✿ x 1 ∧ ( x 2 ∨ x 3 ) ✳ ◮ ❆ ✢♦✇❡r ✐s ❛ ❜♦❞② ♦❜t❛✐♥❡❞ ❜② ❡✈❛❧✉❛t✐♥❣ ❛ ❧❛tt✐❝❡ ♣♦❧②♥♦♠✐❛❧ f ( x 1 , . . . , x k ) ♦♥ s♦♠❡ ❜❛❧❧s x i = B i ✇✐t❤ ♦♣❡r❛t✐♦♥s ∨ = ∪ ✱ ∧ = ∩ ✳ ◮ ❚❤❡ ♣♦✇❡r ♦❢ ❛ ♣♦✐♥t p ✇✐t❤ r❡s♣❡❝t t♦ ❛ ❜❛❧❧ B = B ( c , r ) ✐s P B ( p ) = � p − c � 2 − r 2 ✳ ◮ ❚❤❡ ✭tr✉♥❝❛t❡❞✮ ❉✐r✐❝❤❧❡t✕❱♦r♦♥♦✐ ❝❡❧❧ ♦❢ t❤❡ ❜❛❧❧ B i ✐♥ t❤❡ ✢♦✇❡r f ( B 1 , . . . , B k ) ✐s t❤❡ s❡t C i = { x : f ( P B 1 ( x ) , . . . , f ( P B k ( x )) = f ( P B i ( x ) } , ✇❤❡r❡ f ✐s ❡✈❛❧✉❛t❡❞ ♦♥ t❤❡ ♣♦✇❡rs ✇✐t❤ ♦♣❡r❛t✐♦♥s ∨ = min ✱ ∧ = max ✳ ◮ ❚❤❡ ✇❛❧❧ W ij ❜❡t✇❡❡♥ t❤❡ ❉✐r✐❝❤❧❡t✕❱♦r♦♥♦✐ ❝❡❧❧s C i ❛♥❞ C j ✐❢ W ij = C i ∩ C j ✳

❉❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❱♦❧✉♠❡ ♦❢ ❋❧♦✇❡rs ❚❤❡♦r❡♠ ❙✉♣♣♦s❡ t❤❛t ❡❛❝❤ ✈❛r✐❛❜❧❡ x i ♦❝❝✉rs ✐♥ t❤❡ ❧❛tt✐❝❡ ♣♦❧②♥♦♠✐❛❧ f ( x 1 , . . . , x k ) ❡①❛❝t❧② ♦♥❝❡✳ ▲❡t ǫ ij ❜❡ 1 ✐❢ t❤❡ t❤❡ s❤♦rt❡st s✉❜❡①♣r❡ss✐♦♥ ♦❢ f t❤❛t ❝♦♥t❛✐♥s ❜♦t❤ x i ❛♥❞ x j ✐s ❛ ∨ ♦❢ s❤♦rt❡r ❧❛tt✐❝❡ ♣♦❧②♥♦♠✐❛❧s✱ − 1 ♦t❤❡r✇✐s❡✳ ■❢ t❤❡ ❜❛❧❧ B i ( t ) = B ( c i ( t ) , r i ) ♠♦✈❡ ✐♥ ❛ ❞✐✛❡r❡♥t✐❛❜❧❡ ✇❛②✱ t❤❡♥ t❤❡ ✈♦❧✉♠❡ V ( t ) ♦❢ t❤❡ ✢♦✇❡r f ( B 1 ( t ) , . . . , B k ( t )) ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t ❡❛❝❤ t ❛t ✇❤✐❝❤ t❤❡ ❜❛❧❧s B i ( t ) ❛r❡ ❞✐✛❡r❡♥t ❛♥❞ V ′ = � ǫ ij V ol n − 1 ( W ij ) d ′ ij , 1 ≤ i<j ≤ k ✇❤❡r❡ d ij = � c i − c j � ✳ ❖❜s❡r✈❛t✐♦♥ ✭❘✳ ❈♦♥♥❡❧❧②✮ ■❢ ❛ ❝♦♥✜❣✉r❛t✐♦♥ ♦❢ ❜❛❧❧s ♠❛①✐♠✐③❡s t❤❡ ✈♦❧✉♠❡ ♦❢ ❛ ✢♦✇❡r✱ t❤❡♥ t❤❡ t❡♥s❡❣r✐t② ♦❜t❛✐♥❡❞ ❜② ❝♦♥♥❡❝t✐♥❣ c i ❛♥❞ c j ❜② ❛ ❝❛❜❧❡ ✐❢ ǫ ij = − 1 ❛♥❞ ❛ str✉t ✐❢ ǫ ij = 1 ✐s r✐❣✐❞✳ ❚❤❡ ✈♦❧✉♠❡s ♦❢ t❤❡ ✇❛❧❧s ♣r♦✈✐❞❡ ❛ s❡❧❢ str❡ss V ol n − 1 ( W ij ) ω ij = ǫ ij ✳ d ij

❋♦r♠✉❧❛❡ ✐♥ t❤❡ ❊✉❝❧✐❞❡❛♥ P❧❛♥❡ ◮ ❖r✐❡♥t t❤❡ ♣❧❛♥❡ ❛♥❞ ❛❧❧ t❤❡ ❝✐r❝❧❡s ✐♥ t❤❡ ♣♦s✐t✐✈❡ ❞✐r❡❝t✐♦♥✳ ◮ J ✕ r♦t❛t✐♦♥ ❜② + π 2 ✳ ◮ ❋♦r ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ k ❞✐s❦s D i = D ( c i , r i ) ✇✐t❤ ❜♦✉♥❞❛r② ❝✐r❝❧❡s C i = S ( c i , r i ) ✱ ❞❡♥♦t❡ ❜② p ij t❤❡ ♣♦✐♥t ✇❤❡r❡ C i ❡♥t❡rs C j ✱ ♣r♦✈✐❞❡❞ t❤❛t t❤✐s ♣♦✐♥t ❡①✐sts✳ ◮ ❋♦r ❛ ✢♦✇❡r f ( D 1 , . . . , D k ) ✱ ❞❡✜♥❡ t❤❡ ✈❡rt❡① s❡t V ❛s t❤❡ s❡t ♦❢ ❝r♦ss✐♥❣s p ij ✱ ❧②✐♥❣ ♦♥ t❤❡ ❜♦✉♥❞❛r② ♦❢ t❤❡ ✢♦✇❡r✳ ❚❤❡♦r❡♠ ❆ss✉♠❡ t❤❛t t❤❡ ❞✐s❦s D i = D ( c i , r i ) ♠♦✈❡ s♠♦♦t❤❧② ❛♥❞ t❤❡ s♣❡❡❞ ✈❡❝t♦rs ♦❢ t❤❡✐r ❝❡♥t❡rs ❛r❡ v i ✱ i = 1 , . . . , k ✳ ■❢ ❛❧❧ t❤❡ ❞✐s❦s ❛r❡ ❞✐✛❡r❡♥t✱ t❤❡♥ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❛r❡❛ V ♦❢ t❤❡ ✢♦✇❡r f ( D 1 , . . . , D k ) ❡①✐sts ❛♥❞ ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s ❢♦❧❧♦✇s k V ′ = � �� � � � � � v i , J ǫ ij p ij − ǫ ij p ji = ǫ ij � v i − v j , J p ij � . i =1 p ij ∈V p ji ∈V p ij

❋♦r♠✉❧❛❡ ✐♥ t❤❡ ❊✉❝❧✐❞❡❛♥ P❧❛♥❡ ❈r✐t✐❝❛❧ ❈♦♥✜❣✉r❛t✐♦♥s ❛♥❞ t❤❡ ❍❡ss✐❛♥ ◮ ●✐✈❡♥ ❛ ❧❛tt✐❝❡ ♣♦❧②♥♦♠✐❛❧ f ( x 1 , . . . , x k ) ✱ ❛♥ ❛rr❛♥❣❡♠❡♥t ♦❢ ❞✐s❦s D 1 , . . . , D k ✐s ❝❛❧❧❡❞ ❝r✐t✐❝❛❧ ❝♦♥✜❣✉r❛t✐♦♥ ✐❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❛r❡❛ ♦❢ f ( D 1 , . . . , D k ) ✐s 0 ❢♦r ❛♥② s♠♦♦t❤ ✈❛r✐❛t✐♦♥ ♦❢ t❤❡ ❞✐s❦s✳ ❈♦r♦❧❧❛r② ❆ ❝♦♥✜❣✉r❛t✐♦♥ ♦❢ ❞✐s❦s ✐s ❝r✐t✐❝❛❧ ❢♦r ❛ ❣✐✈❡♥ f ✱ ✐❢ ❛♥❞ ♦♥❧② ✐❢ � � ∀ i : ǫ ij p ij − ǫ ij p ji = 0 . p ij ∈V p ji ∈V ❚❤❡♦r❡♠ ■❢ ❢♦r ❛ ❣✐✈❡♥ ❝r✐t✐❝❛❧ ❝♦♥✜❣✉r❛t✐♦♥✱ t❤❡r❡ ❛r❡ ♥♦ t❛♥❣❡♥t ♣❛✐r ♦❢ ❝✐r❝❧❡s✱ t❤❡ ❝♦♥t❛❝t ♣♦✐♥t ♦❢ ✇❤✐❝❤ ✐s ♦♥ t❤❡ ❜♦✉♥❞❛r② ♦❢ t❤❡ ✢♦✇❡r✱ t❤❡♥ t❤❡ ❛r❡❛ ♦❢ t❤❡ ✢♦✇❡r ✐s t✇✐❝❡ ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t t❤✐s ❝♦♥✜❣✉r❛t✐♦♥ ❛♥❞ ✐ts ❍❡ss✐❛♥ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❜② t❤❡ ❢♦r♠✉❧❛ ǫ ij ( v i − v j ) T ( c i − p ij )( c j − p ij ) T ( v i − v j ) , � Hess ( V , V ) = r i r j sin θ ij p ij ∈V ✇❤❡r❡ ✳

❈♦✈❡r✐♥❣ t❤❡ ▼♦st ✇✐t❤ ✷ ❈♦♥❣r✉❡♥t ❉✐s❦s

❖♣t✐♠❛❧ ❈♦✈❡r✐♥❣s ❜② ✸ ❉✐s❦s ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❇✳ ❙③❛❧❦❛✐ ❋✐rst ❙t❡♣✳ ❈❧❛ss✐❢② ❛❧❧ ❝♦♠❜✐♥❛t♦r✐❛❧❧② ❞✐✛❡r❡♥t ❛rr❛♥❣❡♠❡♥ts t❤❛t ❝❛♥ ♣♦ss✐❜❧② ❣✐✈❡ ❝r✐t✐❝❛❧ ❝♦♥✜❣✉r❛t✐♦♥s✳ ◮ ❋✐♥❞ ❝r✐t❡r✐❛ t❤❛t r✉❧❡ ♦✉t ❣❡♦♠❡tr✐❝❛❧❧② ♥♦t r❡❛❧✐③❛❜❧❡ ❝♦♠❜✐♥❛t♦r✐❛❧ str✉❝t✉r❡s✳ ◮ ❋✐♥❞ ❝r✐t❡r✐❛ t❤❛t ❛r❡ ♥❡❝❡ss❛r✐❧② s❛t✐s✜❡❞ ❜② ♦♣t✐♠❛❧ ❝♦♥✜❣✉r❛t✐♦♥s✳ ◮ ❉❡✈❡❧♦♣ ❛ s♦❢t✇❛r❡ t❤❛t ❧✐sts ❛❧❧ t❤❡ r❡♠❛✐♥✐♥❣ ❝❛s❡s✳ ❚❤✐s ♣r♦❞✉❝❡s ❛ ❧✐st ♦❢ ❝♦♠❜✐♥❛t♦r✐❛❧ ❝♦♥✜❣✉r❛t✐♦♥ t②♣❡s t♦ ❜❡ ❞❡❛❧t ✇✐t❤✳