DOMAINS WHICH ARE LOCALLY UNIFORMLY LINEARLY CONVEX IN THE KOBAYASHI DISTANCE

Recently, in [1], it has been proved that if B is an open unit ball in a Cartesian product l2 × l2 furnished with the lp-norm ‖ · ‖ and kB is the Kobayashi distance on B, then the metric space (B,kB) is locally uniformly convex in linear sense. Our construction of domains, which are locally uniformly convex in their Kobayashi distances, is based on the ideas from [1]. Such domains play an important role in the fixed-point theory of holomorphic mappings (see [1, 2, 4, 13, 14]). In Section 4, we show connections between norm and Kobayashi distance properties.


Introduction
Recently, in [1], it has been proved that if B is an open unit ball in a Cartesian product l 2 × l 2 furnished with the l p -norm • and k B is the Kobayashi distance on B, then the metric space (B,k B ) is locally uniformly convex in linear sense.Our construction of domains, which are locally uniformly convex in their Kobayashi distances, is based on the ideas from [1].Such domains play an important role in the fixed-point theory of holomorphic mappings (see [1,2,4,13,14]).
In Section 4, we show connections between norm and Kobayashi distance properties.

Preliminaries
Throughout this paper, all Banach spaces X will be complex and reflexive, all domains D ⊂ X bounded and convex, and k D will denote the Kobayashi distance on D [6,7,9,10,11,12].
We will use the notions and notations from [2].Here, we recall a few facts only.
The Kobayashi distance k D is locally equivalent to the norm • [9].Indeed, if dist • (x,∂D) denotes the distance in (X, • ) between the point x and the boundary ∂D of the domain D, and diam for all x, y ∈ D and Each open (closed) k D -ball in the metric space (D,k D ) is convex [15] and if D is strictly convex, then every k D -ball is also strictly convex in a linear sense [3,18] (see also [17]).
The metric space (D,k D ) is called a locally uniformly linearly convex space [2] if there exist w ∈ D and the function The open unit ball B H in a Hilbert space is called the Hilbert ball [5,7,8,14,16].
For more useful properties of the Kobayashi distance, see [14].

Examples of locally uniformly linearly convex domains
The first known domain is the Hilbert ball [13,14].Other examples are given in [1].Namely, if B is the open unit ball in a Cartesian product l 2 × l 2 furnished with the l p -norm, where 1 < p < ∞ and p = 2, then the metric space (B,k B ) is also locally uniformly linearly convex.Before stating our main result, we prove the following auxiliary lemma.
Lemma 3.1.Let X be a finite-dimensional Banach space and D a bounded, closed, and strictly convex domain in X.Then, the metric space (D,k D ) is locally uniformly linearly convex.

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Proof.Since D is a bounded and strictly convex domain in X, each k D -ball is strictly convex in a linear sense.Therefore, using the equivalent definition of the k D -boundedness and the compactness argument, we see that the metric space (D,k D ) is locally uniformly linearly convex.Now, we state the main result of this paper.
Theorem 3.2.Let Y be a finite-dimensional subspace of a complex reflexive Banach space X and D a bounded strictly convex domain in X. Suppose that (i) there exists a point x 0 ∈ D 0 = D ∩ Y , (ii) there exists a holomorphic retraction r : D → D 0 , (iii) for every R > 0 and for any three points x, y, and z in the closed k D -ball B(x 0 ,R), there exists a biholomorphic affine mapping T : Then, the metric space (D,k D ) is locally uniformly linearly convex.
Proof.First, observe that D 0 is a strictly convex domain in Y and by (ii), for all u,w ∈ D 0 .This (combined with assumption (i)) implies that the closed Let x, y, and z be three arbitrarily chosen points in the closed k D -ball B(x 0 ,R).By assumption (iii), there exists a biholomorphic affine mapping T : D → D such that Tx,T y,Tz ∈ Y ∩ D 0 and Tx 0 = x 0 .Since this biholomorphic mapping is always a k D -isometry [6, 7, 9, 10, 14], we get

Tx,T y,Tz
Therefore, we may restrict our further considerations to the finite-dimensional Banach space Y .By Lemma 3.1, the metric space (D 0 ,k D0 ) is locally uniformly linearly convex and this implies the same property of (D,k D ).
furnished with the l p -norm, where 1 < p < ∞, and in C n we have a strictly convex norm (i.e., the open unit ball in this norm is strictly convex), then the metric space (B,k B ) is locally uniformly linearly convex.
Example 3.4.In the Cartesian product X = l 2 × l 2 × l 2 , we have the following norm: where 1 < p,q < ∞, p, q = 2, p = q, and (x 1 ,x 2 ,x 3 ) ∈ X.Let B be the open unit ball in X.The metric space (B,k B ) is locally uniformly linearly convex.The proof of this fact is similar to that given in Example 3.3.
Example 3.5.Let X be the Hilbert space l 2 with the standard orthonormal basis {e 1 ,e 2 ,...}.Let D 0 be an arbitrary bounded strictly convex domain in lin{e 1 }.
Let ∂D 0 denote the boundary of D 0 in lin{e 1 }.A strictly convex domain D ∈ X, generated by D 0 , is defined as follows: It is easy to check that we may apply Theorem 3.2, and therefore the metric space (D,k D ) is locally uniformly linearly convex.
Remark 3.6.A construction of more complicated examples is obvious.

Connections between norm and Kobayashi distance properties
There is some connection between the local uniform convexity in linear sense of the unit ball (B,k B ) and the uniform convexity of the whole Banach space.Namely, the following theorem is valid.
Theorem 4.1.Let (X, • ) be a complex Banach space and B the open unit ball in (X, • ).If (B,k B ) is locally uniformly convex in linear sense, then the Banach space (X, • ) is uniformly convex.

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Proof.It is sufficient to show that the ball B(0,1/2) in (X, • ) is uniformly convex.Let We know that the norm • and the Kobayashi distance are locally equivalent and, additionally, we have Hence, by the local uniform convexity in linear sense of the unit ball (B,k B ), we get and therefore It is worth recalling here two facts about strict convexity.As we mentioned in Section 2, the strict convexity of the domain D implies that every k D -ball is also strictly convex in a linear sense [3,18] (see also [17]).It is natural to ask whether the strict convexity of (D,k D ) implies the strict convexity of D. The answer is, no, as the following example shows.