{'fraction_non_alphanumeric': 0.06539735099337748, 'fraction_numerical': 0.011589403973509934, 'mean_word_length': 3.7578740157480315, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
A Characterization of Metric Projection in CAT(0) Spaces

In this paper, we present a characterization of metric projection in CAT(0) spaces by using the concept of quasilinearization. Furthermore, some basic properties of matric projection are investigated.


Introduction
A metric space (X, d) is a CAT(0) space if it is geodesically connected and if every geodesic triangle in X is at least as thin as its comparison triangle in the Euclidean plane. For other equivalent definitions and basic properties, we refer the reader to standard texts such as [1]. Complete CAT(0) spaces are often called Hadamard spaces. Let x, y ∈ X. We write λx ⊕ (1 − λ)y for the the unique point z in the geodesic segment joining from x to y such that d(z, x) = (1 − λ)d(x, y) and d(z, y) = λd(x, y).
We also denote by [x, y] the geodesic segment joining from x to y, that is, Berg and Nikolaev in [2] have introduced the concept of quasilinearization. Let us formally denote a pair (a, b) ∈ X × X by − → ab and call it a vector. Then quasilinearization is the map We say that X satisfies the Cauchy-Schwarz inequality if for all a, b, c, d ∈ X. It known [2, Corollary 3] that a geodesically connected metric space is CAT(0) space if and only if it satisfies the Cauchy-Schwarz inequality. We need the following lemma in the sequel.

Main results
Let C be a nonempty complete convex subset of a CAT(0) space X. It is known [1, Proposition 2.4] that for any x ∈ X there exists a unique point x0 ∈ C such that The mapping PC : X → C defined by PC x = x0 is called the metric projection from X onto C.
We need the following useful lemma to prove our main result.
Proof. Using (1.1) and (1.3), we have which is the desired inequality. This together with Cauchy-Schwarz inequality implies that That is, d(x, u) ≤ d(x, y) for all y ∈ C and so u = PCx. Conversely, let u = PC x. Since C is convex, then z = λy ⊕ (1 − λ)u ∈ C for all y ∈ C and λ ∈ (0, 1). Thus, d(x, u) ≤ d(x, z).
On the other hand, by using Lemma 2.
It is clear that every firmly nonexpansive mapping is monotone. Also, it follows from Cauchy-Schwarz inequality that every firmly nonexpansive mapping is nonexpansive.
Proposition 2.4. Let C be a nonempty closed convex subset of a Hadamard space X. Then, the metric projection PC : X → C ⊆ X is firmly nonexpansive and so it is monotone and nonexpansive.
Proof. Let x, y ∈ X. Since PC x, PC y ∈ C, it follows from Theorem 2.