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  1. ❚❤❡ ❞✐st❛♥❝❡ ❢✉♥❝t✐♦♥ ❢r♦♠ ❛ r❡❛❧ ❛❧❣❡❜r❛✐❝ ✈❛r✐❡t②✳ ◆♦♥❧✐♥❡❛r ❆❧❣❡❜r❛ ✐♥ ❆♣♣❧✐❝❛t✐♦♥s ◆♦✈❡♠❜❡r ✶✹✱ ✷✵✶✽ ●✐♦r❣✐♦ ❖tt❛✈✐❛♥✐ ❯♥✐✈❡rs✐t② ♦❢ ❋❧♦r❡♥❝❡✱ ■t❛❧② ●✐♦r❣✐♦ ❖tt❛✈✐❛♥✐ ❚❤❡ ❞✐st❛♥❝❡ ❢✉♥❝t✐♦♥ ❢r♦♠ ❛♥ ❛❧❣❡❜r❛✐❝ ✈❛r✐❡t②✳ ✶ ✴ ✷✾

  2. P❖❊▼❆ ❛❞✈❡rt✐s❡♠❡♥t ✶✺ P❤❉ P♦s✐t✐♦♥s t❤r♦✉❣❤ P❖❊▼❆✱ ▼❛r✐❡ ❙❦➟♦❞♦✇s❦❛✲❈✉r✐❡ ■❚◆ ✭✷✵✶✾✲✷✵✷✷✮✱ st❛rt✐♥❣ ❢r♦♠ ❙❡♣t❡♠❜❡r ✷✵✶✾✳ ◆❡t✇♦r❦ ♣❛rt♥❡rs✿ ■♥r✐❛✱ ❙♦♣❤✐❛ ❆♥t✐♣♦❧✐s✱ ❋r❛♥❝❡✱ ✭❇❡r♥❛r❞ ▼♦✉rr❛✐♥✮ ❈◆❘❙✱ ▲❆❆❙✱ ❚♦✉❧♦✉s❡ ❋r❛♥❝❡ ✭❉✐❞✐❡r ❍❡♥r✐♦♥✮ ❙♦r❜♦♥♥❡ ❯♥✐✈❡rs✐té✱ P❛r✐s✱ ❋r❛♥❝❡ ✭▼♦❤❛❜ ❙❛❢❡② ❡❧ ❉✐♥✮ ◆❲❖✲■✴❈❲■✱ ❆♠st❡r❞❛♠✱ t❤❡ ◆❡t❤❡r❧❛♥❞s ✭▼♦♥✐q✉❡ ▲❛✉r❡♥t✮ ❯♥✐✈✳ ❚✐❧❜✉r❣✱ t❤❡ ◆❡t❤❡r❧❛♥❞s ✭❊t✐❡♥♥❡ ❞❡ ❑❧❡r❦✮ ❯♥✐✈✳ ❑♦♥st❛♥③✱ ●❡r♠❛♥② ✭▼❛r❦✉s ❙❝❤✇❡✐❣❤♦❢❡r✮ ❯♥✐✈✳ ♦❢ ❋❧♦r❡♥❝❡✱ ■t❛❧② ✭●✐♦r❣✐♦ ❖tt❛✈✐❛♥✐✮ ❯♥✐✈✳ ♦❢ ❇✐r♠✐♥❣❤❛♠✱ ❯❑ ✭▼✐❦❛❧ ❑♦❝✈❛r❛✮ ❋r✐❡❞r✐❝❤✲❆❧❡①❛♥❞❡r✲❯♥✐✈✳ ❊r❧❛♥❣❡♥✱ ●❡r♠❛♥② ✭▼✐❝❤❛❡❧ ❙t✐♥❣❧✮ ❯♥✐✈✳ ♦❢ ❚r♦♠s♦❡✱ ◆♦r✇❛② ✭❈♦r❞✐❛♥ ❘✐❡♥❡r✮ ❆rt❡❧②s ❙❆✱ P❛r✐s✱ ❋r❛♥❝❡ ✭❆r♥❛✉❞ ❘❡♥❛✉❞✮ ❙❡❡ ❤tt♣s✿✴✴❡❛s②❝❤❛✐r✳♦r❣✴❝❢♣✴P❖❊▼❆✲✶✾✲✷✷ ❢♦r ❞❡t❛✐❧s✳ ■♥t❡r❡st❡❞ str♦♥❣ ❝❛♥❞✐❞❛t❡s t♦ t❤❡ P❤❉ ♣♦s✐t✐♦♥ ✐♥ ❋❧♦r❡♥❝❡ ❛r❡ ✐♥✈✐t❡❞ t♦ ❝♦♥t❛❝t ♠❡✱ s❡♥❞✐♥❣ t❤❡✐r ❈❱✳ ●✐♦r❣✐♦ ❖tt❛✈✐❛♥✐ ❚❤❡ ❞✐st❛♥❝❡ ❢✉♥❝t✐♦♥ ❢r♦♠ ❛♥ ❛❧❣❡❜r❛✐❝ ✈❛r✐❡t②✳ ✷ ✴ ✷✾

  3. ■♥ r❡❛❧ ✇♦r❧❞✱ ❧❡✈❡❧ s❡ts ❛r❡ ♦✛s❡t ✈❛r✐❡t✐❡s✱ t❤❡② ❤❛✈❡ str✐❦✐♥❣ ❡♥❣✐♥❡❡r✐♥❣ ❛♣♣❧✐❝❛t✐♦♥s✱ ✐♥ ❈❆❉✴❈❆▼ ♠❛♥✐❢❛❝t✉r✐♥❣ t♦♦❧s✳ ❚❤❡ ❞✐st❛♥❝❡ ❢✉♥❝t✐♦♥ ▲❡t X ⊂ R n ❜❡ ❛♥ ❛❧❣❡❜r❛✐❝ ✈❛r✐❡t②✳ ❚❤❡ ❞✐st❛♥❝❡ ❢✉♥❝t✐♦♥ ❢r♦♠ u ∈ R n t♦ X ✐s d X ( u ) := ♠✐♥ x ∈ X | u − x | ❊✉❝❧✐❞❡❛♥ ❉✐st❛♥❝❡✳ ❇❛s✐❝ q✉❡st✐♦♥✿ ●✐✈❡♥ X ✱ d X ( u ) = ❄✱ ❧❡✈❡❧ s❡ts ❄✱ ♣r♦♣❡rt✐❡s✱ . . . ●✐♦r❣✐♦ ❖tt❛✈✐❛♥✐ ❚❤❡ ❞✐st❛♥❝❡ ❢✉♥❝t✐♦♥ ❢r♦♠ ❛♥ ❛❧❣❡❜r❛✐❝ ✈❛r✐❡t②✳ ✸ ✴ ✷✾

  4. ❚❤❡ ❞✐st❛♥❝❡ ❢✉♥❝t✐♦♥ ▲❡t X ⊂ R n ❜❡ ❛♥ ❛❧❣❡❜r❛✐❝ ✈❛r✐❡t②✳ ❚❤❡ ❞✐st❛♥❝❡ ❢✉♥❝t✐♦♥ ❢r♦♠ u ∈ R n t♦ X ✐s d X ( u ) := ♠✐♥ x ∈ X | u − x | ❊✉❝❧✐❞❡❛♥ ❉✐st❛♥❝❡✳ ❇❛s✐❝ q✉❡st✐♦♥✿ ●✐✈❡♥ X ✱ d X ( u ) = ❄✱ ❧❡✈❡❧ s❡ts ❄✱ ♣r♦♣❡rt✐❡s✱ . . . ■♥ r❡❛❧ ✇♦r❧❞✱ ❧❡✈❡❧ s❡ts ❛r❡ ♦✛s❡t ✈❛r✐❡t✐❡s✱ t❤❡② ❤❛✈❡ str✐❦✐♥❣ ❡♥❣✐♥❡❡r✐♥❣ ❛♣♣❧✐❝❛t✐♦♥s✱ ✐♥ ❈❆❉✴❈❆▼ ♠❛♥✐❢❛❝t✉r✐♥❣ t♦♦❧s✳ ●✐♦r❣✐♦ ❖tt❛✈✐❛♥✐ ❚❤❡ ❞✐st❛♥❝❡ ❢✉♥❝t✐♦♥ ❢r♦♠ ❛♥ ❛❧❣❡❜r❛✐❝ ✈❛r✐❡t②✳ ✸ ✴ ✷✾

  5. ▲❡✈❡❧ ❝✉r✈❡s ♦❢ t❤✐s ✏s♦♠❜r❡r♦✑ ❛r❡ r❡❛❧ ♦❝t✐❝s✱ ✇✐t❤ t✇♦ ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥ts✳ ❇② ♠♦✈✐♥❣ ❛❝r♦ss t❤❡ s✐♥❣✉❧❛r s❡t✱ t❤❡ ♣♦✐♥t ♦♥ t❤❡ ❡❧❧✐♣s❡ ✇❤✐❝❤ r❡❛❧✐③❡s t❤❡ ♠✐♥✐♠✉♠ ❞✐st❛♥❝❡ ❢r♦♠ ❥✉♠♣s✳ ■t r❡♠❛✐♥s ✇❡❧❧ ❞❡✜♥❡❞✱ ❜② ❝♦♥t✐♥✉❛t✐♦♥✱ ❛s ❛ ♣♦✐♥t ✇❤♦s❡ ♥♦r♠❛❧ ♠❡❡ts ✳ ●r❛♣❤✐❝ ♦❢ t❤❡ ❞✐st❛♥❝❡ ❢✉♥❝t✐♦♥ ❢r♦♠ ❛ ❡❧❧✐♣s❡ ✭❝♦✉rt❡s② ❜② ❆♥t♦♥✐♦ ●r❡❝♦✮ ●✐♦r❣✐♦ ❖tt❛✈✐❛♥✐ ❚❤❡ ❞✐st❛♥❝❡ ❢✉♥❝t✐♦♥ ❢r♦♠ ❛♥ ❛❧❣❡❜r❛✐❝ ✈❛r✐❡t②✳ ✹ ✴ ✷✾

  6. ●r❛♣❤✐❝ ♦❢ t❤❡ ❞✐st❛♥❝❡ ❢✉♥❝t✐♦♥ ❢r♦♠ ❛ ❡❧❧✐♣s❡ ✭❝♦✉rt❡s② ❜② ❆♥t♦♥✐♦ ●r❡❝♦✮ ▲❡✈❡❧ ❝✉r✈❡s ♦❢ t❤✐s ✏s♦♠❜r❡r♦✑ ❛r❡ r❡❛❧ ♦❝t✐❝s✱ ✇✐t❤ t✇♦ ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥ts✳ ❇② ♠♦✈✐♥❣ P ❛❝r♦ss t❤❡ s✐♥❣✉❧❛r s❡t✱ t❤❡ ♣♦✐♥t ♦♥ t❤❡ ❡❧❧✐♣s❡ ✇❤✐❝❤ r❡❛❧✐③❡s t❤❡ ♠✐♥✐♠✉♠ ❞✐st❛♥❝❡ ❢r♦♠ P ❥✉♠♣s✳ ■t r❡♠❛✐♥s ✇❡❧❧ ❞❡✜♥❡❞✱ ❜② ❝♦♥t✐♥✉❛t✐♦♥✱ ❛s ❛ ♣♦✐♥t ✇❤♦s❡ ♥♦r♠❛❧ ♠❡❡ts P ✳ ●✐♦r❣✐♦ ❖tt❛✈✐❛♥✐ ❚❤❡ ❞✐st❛♥❝❡ ❢✉♥❝t✐♦♥ ❢r♦♠ ❛♥ ❛❧❣❡❜r❛✐❝ ✈❛r✐❡t②✳ ✹ ✴ ✷✾

  7. ❚❤❡ ❝r✐t✐❝❛❧ ♣♦✐♥ts ♦❢ t❤❡ ❞✐st❛♥❝❡ ❢✉♥❝t✐♦♥ ❢r♦♠ u ♦♥ X ■❢ X ✐s ❛ s♠♦♦t❤ s✉❜✈❛r✐❡t②✱ t❤❡ ♠✐♥✐♠✉♠ ♦❢ t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ u ✐s ❛tt❛✐♥❡❞ ❛♠♦♥❣ t❤❡ ♣♦✐♥ts x s✉❝❤ t❤❛t T x X ⊥ ( u − x ) ✳ ❚❤❡s❡ ❛r❡ t❤❡ ❝r✐t✐❝❛❧ ♣♦✐♥ts ♦♥ X ♦❢ t❤❡ ❞✐st❛♥❝❡ ❢✉♥❝t✐♦♥ ❢r♦♠ u ✳ ❈❤❡❝❦✐♥❣ ❛❧❧ ♦❢ t❤❡ ❝r✐t✐❝❛❧ ♣♦✐♥ts ❣✉❛r❛♥t❡❡s t♦ ❝♦♠♣✉t❡ t❤❡ ❣❧♦❜❛❧ ❞✐st❛♥❝❡ ❢r♦♠ u t♦ X ✳ X x u T x X ●✐♦r❣✐♦ ❖tt❛✈✐❛♥✐ ❚❤❡ ❞✐st❛♥❝❡ ❢✉♥❝t✐♦♥ ❢r♦♠ ❛♥ ❛❧❣❡❜r❛✐❝ ✈❛r✐❡t②✳ ✺ ✴ ✷✾

  8. ✷ ♥♦r♠❛❧ ❧✐♥❡s✱ ♦♥❧② ✶ r❡❛❧✐③❡s t❤❡ ♠✐♥✐♠✉♠✳ ✹ ♥♦r♠❛❧ ❧✐♥❡s✱ ♦♥❧② ✶ r❡❛❧✐③❡s t❤❡ ♠✐♥✐♠✉♠✳ ■♥ t❤❡ ✜rst ❝❛s❡ t❤❡r❡ ❛r❡ ♦t❤❡r ✷ ❝♦♠♣❧❡① ❝♦♥❥✉❣❛t❡s ♥♦r♠❛❧ ❧✐♥❡s✱ ✹ ✐s t❤❡ ❡①♣❡❝t❡❞ ❛♥❞ ❣❡♥❡r❛❧ ✈❛❧✉❡ ✭ ❊❉❞❡❣r❡❡ ✭❡❧❧✐♣s❡✮ ✹✮✳ ❚❤❡ ♥♦r♠❛❧ ❧✐♥❡s ❢r♦♠ ❛ ♣♦✐♥t t♦ ❛ ❡❧❧✐♣s❡✳ P❤✐❧♦s♦♣❤②✿ t❤❡ ♣♦✐♥t r❡❛❧✐③✐♥❣ t❤❡ ♠✐♥✐♠✉♠ ❞✐st❛♥❝❡ ❝❛♥♥♦t ❜❡ ❝♦♥s✐❞❡r❡❞ ❛❧♦♥❡✳ ❆❧❧ t❤❡ ❝r✐t✐❝❛❧ ♣♦✐♥ts ❤❛✈❡ t♦ ❜❡ ❝♦♥s✐❞❡r❡❞ t♦❣❡t❤❡r✳ ●✐♦r❣✐♦ ❖tt❛✈✐❛♥✐ ❚❤❡ ❞✐st❛♥❝❡ ❢✉♥❝t✐♦♥ ❢r♦♠ ❛♥ ❛❧❣❡❜r❛✐❝ ✈❛r✐❡t②✳ ✻ ✴ ✷✾

  9. ✹ ♥♦r♠❛❧ ❧✐♥❡s✱ ♦♥❧② ✶ r❡❛❧✐③❡s t❤❡ ♠✐♥✐♠✉♠✳ ■♥ t❤❡ ✜rst ❝❛s❡ t❤❡r❡ ❛r❡ ♦t❤❡r ✷ ❝♦♠♣❧❡① ❝♦♥❥✉❣❛t❡s ♥♦r♠❛❧ ❧✐♥❡s✱ ✹ ✐s t❤❡ ❡①♣❡❝t❡❞ ❛♥❞ ❣❡♥❡r❛❧ ✈❛❧✉❡ ✭ ❊❉❞❡❣r❡❡ ✭❡❧❧✐♣s❡✮ ✹✮✳ ❚❤❡ ♥♦r♠❛❧ ❧✐♥❡s ❢r♦♠ ❛ ♣♦✐♥t t♦ ❛ ❡❧❧✐♣s❡✳ P❤✐❧♦s♦♣❤②✿ t❤❡ ♣♦✐♥t r❡❛❧✐③✐♥❣ t❤❡ ♠✐♥✐♠✉♠ ❞✐st❛♥❝❡ ❝❛♥♥♦t ❜❡ ❝♦♥s✐❞❡r❡❞ ❛❧♦♥❡✳ ❆❧❧ t❤❡ ❝r✐t✐❝❛❧ ♣♦✐♥ts ❤❛✈❡ t♦ ❜❡ ❝♦♥s✐❞❡r❡❞ t♦❣❡t❤❡r✳ ✷ ♥♦r♠❛❧ ❧✐♥❡s✱ ♦♥❧② ✶ r❡❛❧✐③❡s t❤❡ ♠✐♥✐♠✉♠✳ ●✐♦r❣✐♦ ❖tt❛✈✐❛♥✐ ❚❤❡ ❞✐st❛♥❝❡ ❢✉♥❝t✐♦♥ ❢r♦♠ ❛♥ ❛❧❣❡❜r❛✐❝ ✈❛r✐❡t②✳ ✻ ✴ ✷✾

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