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1 Proof mining in -trees and hyperbolic spaces Laurent iu Leus tean TU Darmstadt, Germany and Institute of Mathematics Simion Stoilow of the Romanian Academy

  1. 1 Proof mining in -trees and hyperbolic spaces Laurent ¸iu Leus ¸tean TU Darmstadt, Germany and Institute of Mathematics ”Simion Stoilow” of the Romanian Academy

  2. ✸ ✘ ✮ � ★ ✧ ✆ ✹ ✸ ✹ ✡ ✎ ✚ ✡ ✆ ★ ✝ ✻ ✰ ✪ ✷ ✱ ✶ ✶ ✶ ✵ ✱ ✎ ✲ ✝ ✺ ✮✯ ✽ ✦ ✁ ✝ ✟ ❀ ✿ ✾ ✻ ✝ ❁ ✘ ★ ✦ ✟ ☛ ✷ ✱ ✶ ✶ ✶ ✵ ✱ ✳ ✰ ★ ❂ ✡ ✄ ☎ ★ ✑ ✝ ✓ ✡ ✝ ✻ ✎ ✝ ✏ ✎ ✌ ✑ ✎ ✏ ✹ ❃ ✎ ✬ ✆ ☎ ✄ ✘ ✂ ✡ ✁ ✚ � ✎ ✡ ✸ ✡ ☛ ✎ ☛ ✫ ✑ ✏ ✓ ✫ ✓ ❂ ✝ ✆ ☎ ✄ ✦ ✆ ✆ ✟ ✝ ✦ ✁ ★ ✩ ✦ � ★ ✧ ✦ ✮ ✩ 2 General metatheorem Theorem 1 (Gerhardy/Kohlenbach, 2005) Polish space, compact metric space, ”small” type, , ✡☞☛✍✌ ✆✞✝✠✟ contain only free, resp. free. Assume that ✆✞✝✒✟ ✡☞✓ ✡☞☛ ✔✖✕ ✛✢✜ ✗✙✘ ✡☞✚ ✣✥✤ ✡☞☛ ✎✞✪ ✆✞✝ ✡☞✓ ✎✭✬ Then there exists a computable functional such ✰✍✱ ✰✴✳ that the following holds in all nonempty metric spaces : for all representatives of and all , if there ✝✼✻ exists an such that , then ✆✞✝ ✎✞✪ ✆✞✝ ✡☞✓ The theorem also holds for nonempty hyperbolic spaces , ✡❅❄ CAT(0)-spaces, normed spaces, inner product spaces.

  3. ✻ � � � � ✁ ✂ � ✁ 3 General metatheorem the metatheorem can be used as a black box: infer new uniform existence results without any proof analysis run the extraction algorithm: extract an explicit effective bound given proof new proof for the stronger result new mathematical proof of a stronger statement which no longer relies at any logical tool

  4. � ✽ 4 Metatheorems for other classes of spaces adapt the metatheorem to other classes of spaces: 1. the language may be extended by -majorizable constants 2. the theory may be extended by purely universal axioms

  5. ✬ ✩ ✩ ✸ ✏ ✑ ✒✓ ✔ ✝ ✡ ✡ ✟ ✧ ✡ � ★ ✘ ✚ ✡ ✌ ✡ ✎ ✙ ✂ ✡ ✆ ✁ ✧ ✎ ✂ ✟ ✘ ✡ ✬ ✘ ✡ ✡ ☛ ☞ � ✩ ✝ ✧ ✆ ✡ ✚ ✡ � ✎ ✝ ✚ ✩ � ✡ ✧ ✎ ✙ ✝ ✆ ✡ ✎ ✡ ✚ ✚ ✆ ✧ ✡ � ✎ ❂ ✕ ✛ ✩ ✘ ✚ ✡ ✧ ✎ ✝ ✚ ✆ ✎ ✩ ✁ ✝ ✁ ✌ ✟ ✡ ✩ ✘ ✎ ☞ ☛ ✡ ✂ ✄ � ✝ ☎ ✎ ✩ ✡ ✆ ✚ ✆ � ✡ ✩ ✆ ✚ ✚ ✩ ✸ ✏ ✡ � ✆ ✘ ✗ ✆ ✕✖ ☛ ✂ ✎ ✩ ✁ ✘ ✎ ✘ ★ ✑✒✓ ✝ � ✡ ✧ ✩ ✡ ✩ � ✡ ✝ ✔ ✆ � defined to be: ✆✞✝ ✡☞✚ Let the Gromov product of is Gromov hyperbolic if it is is metric space . ✁ ✎✍ ✆✞✝ is called Gromov hyperbolic spaces iff for all and ✆✞✝ ✁ ✎✍ ✆✞✝ -hyperbolic for some with respect to the base point ✎ ✞✝ if for all ✆✞✝ ✎ ✠✟ , ✆✞✝ ✎✭✬ . , is 5

  6. ✎ ✝ ✝ ✎ � ✡ ✄ ✎ � ✡ ✩ ✆ ✄ ✚ ✂ ✎ ✚ ✧ ✡ ✝ ✆ ✄ ✚ ✘ ✂ ✛ ✚ ✕ ✂ ✂ ✂ ✄ ✎ ✕ ✡ ✡ ✚ ✡ ✘ ✆ ✣ ✡ ✡ ✚ ✡ ✘ ✗ ✔ ✆ � ❃ ✡ ✂ ✁ ✆ ✂ ✝ ✙ ✎ ✧ ✡ ✩ ❂ ❂ � ✘ ✦ ✗ ✡ ✂ ☞ ✂ ☛ ✡ ✡ ☎ ✝ ✡ ✡ ✣ ✔ ✣ ✜ ✛ ✚ ✡ ✘ ✕ ✄ ✕ ✔ ✧ ✆ � ✄ ✚ ✂ ✝ ✎ ✩ ✡ ✄ ✡ ✚ ❁ ✄ � ✡ ✄ ✧ ✡ ✄ ✩ ✗ 6 Gromov hyperbolic spaces The theory of Gromov hyperbolic spaces, -hyperbolic is ✡☞✚ ✛✢✜ defined by extending as follows: 1. add a constant of type , ✡ ✁� 2. add the axioms ✆✞✝ ✡☞✚ ✆✞✝ Theorem 1 holds also for -hyperbolic and nonempty ✛✙✜ Gromov hyperbolic spaces

  7. ✬ ✎ ✆ ❄ ✆ ✎ ✚ ✆ ❄ ✡ ✧ ✆ ✎ ✩ ✡ � ✎ ✎ ❂ ✆ ☎ ✟ � ✡ � ✚ ✝ ✡ ✩ ✡ ✆ ❄ ☎ ✎ ❄ ✆ ✡ ✟ ✩ ✎ ✄ ❄ ✆ ✩ ✡ ✝ ✡ ☎ ✎ ✡ ✁ ✡ ✡ ✩ ✛ ✯ ✄ ✘ ❄ ✆ ✝ ✩ ✗ ✎ ✯ � ★ ✗ ☞ ✡ ☎ ✛ ✙ ✝ � ✩ ✟ ✎ ✝ � ✚ ✆ ✧ ✡ � ✆ ☎ � ✡ ✎ ✝ ✝ � ✩ ✯ ✄ ❄ ✡ ✩ ✚ ✎ ✄ ✝ ❄ ☎ ✎ ✚ ✆ ✧ ✡ ❄ ✆ ✡ ✘ ✩ ✎ ✎ ❂ � ☎ ✟ � ✎ ✚ ✆ ✪ ✧ ✚ ❄ ✆ ✘ ✡ ✂ ✎ ✆ ✘ ✡ ✎ ✛ ❄ ✯ ✘ ✲ ✘ ✲ ✗ ☞ ✡ ☎ ✆ ✆ ✡ ✡ ❄ ✆ ✝ ✡ ✩ ✟ � ✎ ❄ ✎ ✆ ✄ ✝ ✡ ✩ ✡ ✄ ✂ � ✆ ✚ ✎ ✡ ✝ ✎ ✝ � ✚ ✆ ✆ ✧ ✩ ✎ ✡ ✆ ✂ ❄ ✎ Notation: and A [Takahashi, Goebel/Kirk, Reich/Shafrir, Kohlenbach] -hyperbolic space is a triple ✆✞✝ ✡ ✁� ✡❅❄ W-hyperbolic spaces ✡ ✁� ✡ ✁� ✆✞✝ ✡ ✁� ✡ ✁� s.t. ✡ ✁� ✡❅❄ ✎✭✬ where ✆✞✝ ✆✞✝ is metric space ✎✭✬ 7

  8. ✬ � ✝ ✩ ✁ ✡ ✘ � � ✎ ✘ ✁ ✆ ✡ ✎ ✚ ✘ ✆ ✡ � � ✎ ✘ ✝ ✄ ✧ ✚ ✘ ✛ ✚ ✕ ❂ ✎ � ✡ ✆ ✘ ✚ ✝ ✎ ✩ ✡ ✎ ✚ ✝ ❄ ✚ ✘ ☎ ✡ ✩ ✁ ✪ ✛ ✏ ✡ ✽ ✗ ✯ ✂ ✎ ✄ ✡ ✘ ✆ ✧ � ✎ ✁ � ✙ ✆ ✚ ✟ ✎ ✄ ✄ ✄ ✎ ✎ ☎ ✆ ✂ ✡ ✄ ★ ✆ ✂ ✆ ✚ ✝ ✛ ✏ ✡ ✽ ✗ ✧ 8 -trees -trees introduced by Tits(’77) A geodesic in a metric space is a map s.t. for all ✡☞✚ , ✡ ✆☎ is said to be a geodesic space if every two points are joined by a geodesic. A metric space is an -tree if is a geodesic space containing ✡☞✚ no homeomorphic image of a circle. is an -tree is a 0-hyperbolic geodesic space is a -hyperbolic space satisfying ✆✞✝ ✆✞✝ ✆✞✝

  9. ❂ ✡ ✎ � ✡ ✩ ✆ ✄ ✚ ✂ ✝ ✎ ✧ ✄ ✚ ✚ ✘ ✂ ✛ ✚ ✕ ✂ ❂ ✂ � ✁ ✡ ✄ ✡ ❃ ✣ ✁ ✡ ✚ ✡ ✘ ✗ ✕ ✔ � ✬ ✙ ✡ ✎ ✧ ✡ ✩ ✆ ✄ ✚ ✂ ✝ ✎ � � ✎ ✧ ✜ ✦ ✂ ✁ � ✕ ✆ ✗ ✣ ✛ ✄ ✘ ✡ ✚ ✕ ✔ ✡ ✣ ✜ ✝ ✡ ✁ ✄ ✄ ✚ ✂ ✝ ✎ ✩ ✡ � ✚ ✩ ❁ ✄ ✔ � ✡ ✄ ✧ ✡ ✄ ✛ axiom: Theorem 1 holds also for ✡❅❄ -tree ✆✞✝ ✆✞✝ results from ✡❅❄ -trees ✗✙✘ ✡☞✚ -tree ✡❅❄ ✛✢✜ ✆✞✝ and nonempty by adding the -trees. 9

  10. ✂ ✆ ✟ ☎ ✆ ❂ ✽ ✡ ✩ ✆ ☎ ✝ ✝ ☎ ✎ ✚ ✂ ☎ ✂ ✸ ✄ ✂ ✂ ✁ ✸ � ✡ ✸ ✎ ✆ ✄ ✯ ✡ ✸ ✛ ☎ ✡ ☞ ✆ ✪ ✛ ✡ ✬ ☞ ✆ ✲ ✎ ✡ ☞ ✆ ✯ ✆ � ✎ ☛ ✩ ✆ ★ ☞ ✆ ★ ✡ ✆ ☞ ✛ ✆ ✡ ☞ ✆ � ☎ ✡ ☞ ✾ ✸ ✆ ✎ ✛ ✂ ✡ ✘ ✆ ✡ ✛ ✡ ❂ ✾ ✚ ✸ ❂ ✎ ✽ ✡ ✩ ✆ ✚ ✸ ✎ ★ ✽ ✡ ☞ ✚ � ✘ ★ ✩ ✡ ✝ ✡ ✽ ✆ 10 Uniformly convex W-hyperbolic spaces is uniformly convex if for any , and there exists ✡❅❄ a s. t. for all , ✆✞✝ (1) ✆✞✝ A mapping providing such a for ✡ ✞✝ given and is called a modulus of uniform convexity .

  11. ✎ ✩ ✷ ✪ ✚ ✄ ✆ ✝ ✡ ✎ ✳ ❂ ✂ ✆ ✜ ✟ ❃ ✡ ✝ ✆ ✸ ✎ ✆ ✝ ✩ ✡ ☎ ✄ ✆ ✡ ☎ ✽ ✎ ✾ ✂ ☎ ✟ ✆ ✜ ✦ ✌ ❄ ✚ ✎ � ✔ ✕ ✗ ✘ ✡ ✡ ✡ ✆ ✣ ❄ ✆ ✘ ✡ ✆ � ✎ ✡ ✡ � ✌ ✆ ✆ ✄ ✆ ✸ � ✸ ✎ ✄ ✌ ✆ ✄ ✆ ✡ ✆ ✄ ✡ ✆ ✁ ✆ ✄ ☞ ☞ ☞ ✄ ✂ ❄ ✦ ✸ ✌ ✦ � ✌ ✣ ✡ ✝ ✚ � ✔ ✕ ✗ ✘ ✡ ✡ ✘ ✆ ✣ ❄ ✔ ✕ ✗ ✦ � ✄ ✡ ✸ ✂ ✚ ✡ ✆ ✩ ✽ ✁ ✎ ✁ ✂ ✸ ✂ ✂ ✚ ✂ ✄ ✄ ✚ ✎ ✩ ✽ ✄ ✡ ✡ ✝ ✆ ✽ ✄ ❁ 11 Uniformly convex W-hyperbolic spaces The theory of uniformly convex -hyperbolic spaces ✡❅❄ ✛✢✜ extends the theory as follows: ✡☞✚ ✛✢✜ 1. add a new constant of type , 2. add the following axioms: ✞✠✟ ✎✭✬ Theorem 1 holds also for and nonempty uniformly ✛✢✜ ✡❅❄ convex -hyperbolic spaces ✡☞✚ ✡❅❄

  12. ✬ ✯ ✝ ✁ ✝ ✎ ✁ � ✟ ☎ ✆ ✄ � ✁ ☎ ✁ ✝ ✡ ✝ ✄ ✯ ✌ ✝ ✏ � ✄ � ✝ ☞ ✄ ✎ ✁ ✝ ✄ ✡ ✁ ✚ ✞ ✁ ✝ ✕ ✖ ✆ ✏ ★ ✝ ✁ � ✄ � ✁ ✝ ★ ✡ ✁ � ✛ ✎ ✡ ☞ ✗ ✱ ✂ ✁ ✎ � ✯ ✆ ✘ � ✏ ❄ ✎ ❄ ✡ ✘ ✆ ✄ ☎ ✏ ❂ ✪ ✝ ✄ ✆ ✩ ✚ ✎ ✏ ✄ ✚ ✡ ★ ✩ ✡ ✝ ✩ ✏ ✡ 12 Fixed point theory of nonexpansive mappings -hyperbolic, convex, sequence in ✡☞✚ nonexpansive if for all ✆✞✝ The Krasnoselski-Mann iteration starting from : asymptotic regularity - defined by Browder/Petryshyn(66) for normed spaces: is -asymptotically regular if for all , ✆✞✝

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