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❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ ❞❡ ▼❛r✐♥❣á ✲ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛
❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡ ■♥t❡❣r❛❧✿ ✉♠ ❑■❚ ❞❡ ❙♦❜r❡✈✐✈ê♥❝✐❛
❝ P✉❜❧✐❝❛çã♦ ❡❧❡trô♥✐❝❛ ❞♦ ❑■❚

❤tt♣✿✴✴✇✇✇✳❞♠❛✳✉❡♠✳❜r✴❦✐t

❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡
Pr♦❢✳ ❉♦❤❡rt② ❆♥❞r❛❞❡
❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ ❞❡ ▼❛r✐♥❣á
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ✲ ✽✼✵✷✵✲✾✵✵ ▼❛r✐♥❣á✲P❘✱ ❇r❛③✐❧

❙✉♠ár✐♦

✶✳ Pr❡❧✐♠✐♥❛r❡s

✸

✷✳ ❋✉♥çõ❡s ❈♦♥tí♥✉❛s

✻

✸✳ ❚❡♦r❡♠❛s ❞❡ P♦♥t♦ ❋✐①♦

✾

✹✳ ■♥tr♦❞✉çã♦ ❛s ❊❉❖✬s

✶✵

✺✳ Pr♦✈❛ ❞♦ ❚❡♦r❡♠❛ ❞❡ ❊①✐stê♥❝✐❛

✶✸

✻✳ ❚r❛♥s❢♦r♠❛❞❛ L ❞❡ ▲❛♣❧❛❝❡

✶✽

✼✳ Pr♦♣r✐❡❞❛❞❡s

✷✵

✽✳ ❆ ■♥✈❡rs❛ ❞❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡

✷✹

✾✳ ❋r❛çõ❡s P❛r❝✐❛✐s

✷✹

✶✵✳❚❡♦r❡♠❛ ❞❛ ❈♦♥✈♦❧✉çã♦

✷✼

✶✶✳❆♣❧✐❝❛çõ❡s ❛ ❊❉❖✬s

✷✽

✶✷✳▼ét♦❞♦s ♣❛r❛ ❞❡t❡r♠✐♥❛r ❛ tr❛♥s❢♦r♠❛❞❛ ✐♥✈❡rs❛ ❞❡ ▲❛♣❧❛❝❡ ✷✾
✶✸✳❆♣❧✐❝❛çã♦ ❛ s✐st❡♠❛s ❞❡ ❊❉❖✬s

✸✵

✷

Pr♦❢✳ ❉♦❤❡rt② ❆♥❞r❛❞❡

❘❡s✉♠♦✿ ❊st❛s ♥♦t❛s ❢♦r❛♠ ❡s♣❡❝✐❛❧♠❡♥t❡ ❡❧❛❜♦r❛❞❛s ♣❛r❛ s❡r✈✐r❡♠ ❞❡ t❡①✲ t♦ ♣❛r❛ ♦ ♠✐♥✐✲❝✉rs♦ s♦❜r❡ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡✳

❊st❡ ♠✐♥✐✲❝✉rs♦ é

✐♥tr♦❞✉tór✐♦ ❡ ❡①✐❣❡✲s❡ ♦ ♠í♥✐♠♦ ❞❡ ♣ré✲r❡q✉✐s✐t♦s✳
❆❝♦♠♣❛♥❤❛♠ ❡st❛s ♥♦t❛s ✉♠ ❞✐s❦❡t❡ ❝♦♥t❡♥❞♦ ❛rq✉✐✈♦s ❡♠ ▼❛♣❧❡ ❝♦♠ ♦s
❡①❡♠♣❧♦s ❡ ♦✉tr❛s ❛t✐✈✐❞❛❞❡s ♣❛r❛ s❡r❡♠ r❡❛❧✐③❛❞❛s ♥♦ ❝♦♠♣✉t❛❞♦r✳

❝
✶✳

❑■❚ ✲

✸

❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡ ■♥t❡❣r❛❧

Pr❡❧✐♠✐♥❛r❡s

◆❡st❛ s❡çã♦ ✈❛♠♦s ✈❡r ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❡❧❡♠❡♥t❛r❡s q✉❡ s❡rã♦ út❡✐s ♥♦ r❡s✲ t❛♥t❡ ❞❛ t❡♦r✐❛✳
❯♠ ❡s♣❛ç♦ ♠étr✐❝♦ é ✉♠ ♣❛r (M, d), ♦♥❞❡ M é ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ ❡ d ✉♠❛ ❢✉♥çã♦ d : M × M → R q✉❡ ♣❛r❛ t♦❞♦s ♦s ♣♦♥t♦s x ❡ y ❞❡ M s❛t✐s❢❛③✿
❛✮ d(x, y) ≥ 0, ✭♣♦s✐t✐✈❛✮
❜✮ d(x, y) = 0 ⇐⇒ x = y, ✭♥ã♦ ❞❡❣❡♥❡r❛❞❛✮
❝✮ d(x, y) = d(y, x), ✭s✐♠étr✐❝❛✮
❞✮ d(x, z) ≤ d(x, y) + d(y, z) ✭❞❡s✐❣✳ tr✐❛♥❣✉❧❛r✮✳
❆ ❢✉♥çã♦ d é ❝❤❛♠❛❞❛ ✉♠❛ ♠étr✐❝❛ ❡ d(x, y) s✐❣♥✐✜❝❛ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ x
❡ y✳
❖ ❡s♣❛ç♦ ♠étr✐❝♦ q✉❡ t❡♠♦s ❞❡ ✐♠❡❞✐❛t♦ ❡ ♠❛✐s ✐♥t❡r❡ss❛♥t❡ é ♦ Rn ✱ ❝✉❥❛
♠étr✐❝❛ d : Rn × Rn → R é ❞❛❞❛ ♣♦r d(x, y) = x − y , ♦♥❞❡ x−y =

x − y, x − y .

❊st❛ é✱ ♣♦r r❛③õ❡s ó❜✈✐❛s✱ ❝❤❛♠❛❞❛ ♠étr✐❝❛ ❡✉❝❧✐❞✐❛♥❛✳
P♦❞❡✲s❡ ♣r♦✈❛r q✉❡ ❡♠ Rn t♦❞❛s ❛s ♠étr✐❝❛s sã♦ ❡q✉✐✈❛❧❡♥t❡s✱ ✐st♦ é✱ ❞✉❛s
♠étr✐❝❛s d1 ❡ d2 s♦❜r❡ Rn q✉❛✐sq✉❡ s❛t✐s❢❛③❡♠ d1 (x, y) ≤ c1 d2 (x, y) ≤ c2 d1