Intrinsic quasi-metrics

The point pair function $p_G$ defined in a domain $G\subsetneq\mathbb{R}^n$ is proven to be a quasi-metric with the best possible constant. For a convex domain $G\subsetneq\mathbb{R}^n$, a new intrinsic quasi-metric called the function $w_G$ is introduced. Several sharp results are established for these two quasi-metrics, and their connection to the triangular ratio metric is studied.


Introduction
In geometric function theory, one of the key concepts is an intrinsic distance. This notion means a distance between two points fixed in a domain that not only depends on how close these points are to each other but also takes into account how they are located with respect to the boundary of the domain. A well-known example of an intrinsic metric is the hyperbolic metric [1], which is conformally invariant and therefore has several properties useful in the study of the distortion under different functions.
However, the hyperbolic metric is not the only metric that can be used to model the intrinsic distances and not even its key features are unique. On the contrary, it has several generalizations that share its main properties and whose behaviour and qualities have also been widely studied [5,Ch. 5,. Especially during the past thirty years, various hyperbolic type metrics have been introduced, see [2,5,7,9,10,12].
This often raises the question about the reason for introducing new metrics and studying them instead of just focusing on those already existing. To answer this, it should be first noted that the slightly different definitions of the intrinsic metrics mean that they have unique advantages and suit for diverse purposes. Consequently, new metrics can be used to discover various intricate features of geometric entities that would not be detected with some other metrics. For instance, many hyperbolic type metrics behave slightly differently under quasiconformal mappings and analysing these differences can give us a better understanding of how such mappings really work [11].
Furthermore, new metrics can also bring information about the already existing metrics. Calculating the exact value of the hyperbolic metric in a domain that cannot be mapped onto the unit disk with a conformal mapping is often impossible but we can estimate it by using other intrinsic metrics with simpler definitions [5,Ch. 4.3,. However, in File: main.tex, printed: 2020- 11-5, 1.49 order to do this, we need to know the connection between the different metrics considered and to be able to create upper and lower bounds for them. Finding sharp inequalities for intrinsic metrics can often help us with some related applications and, for instance, in the estimation of condenser capacities [5,Ch. 9,.
Another noteworthy motivation for studying several different metrics is that their inequalities can tell us more about the domain where the metrics are defined. The definition for a uniform domain can be expressed with an inequality between the quasihyperbolic metric and the distance ratio metric as in [5,Def. 6.1,p. 84]. Similarly, some other inequalities can be used to determine whether the domain is, for instance, convex or not, like in Theorem 3.17. Further, Corollary 3.20 even shows an equality between metrics that serves as a condition for when the domain can be mapped onto a half-space with just Euclidean isometries.
In this paper, we consider two different intrinsic quasi-metrics. By a quasi-metric, we mean a function that fulfills all the conditions of a metric otherwise but only a relaxed version of the triangle inequality instead of the inequality itself holds for this function, see Definition 2.1 and the inequality (2.2). The first quasi-metric considered is the point pair function introduced by Chen et al. in 2015 [2], and the other quasi-metric is a function defined for the first time in Definition 4.1 in this paper. We also study the triangular ratio metric introduced by P. Hästö in 2002 [7] for one of the main results of this paper is showing how our new quasi-metric can be used to create a very good lower bound for this metric, especially in the case where the domain is the unit disk.
The structure of this paper is as follows. In Section 3, we prove the exact constant with which the point pair function is a quasi-metric and show how it can be used together with the triangular ratio metric to tell us about the shape of the domain. Then, in Section 4, we introduce a new quasi-metric and also inspect its connection to triangular ratio metric. Finally, in Section 5, we focus on the case where the domain is the unit disk and find several sharp inequalities between different hyperbolic type metric and quasi-metrics considered. Especially, we investigate how the new quasi-metric can be used to estimate the value of the triangular ratio metric in the unit disk, see Theorem 5.7 and Conjecture 5.7.
Acknowledgements. This research continues my work with Professor Matti Vuorinen in [12,13,14]. I am indebted to him for all guidance and other support. My research was also supported by Finnish Concordia Fund.

Preliminaries
In this section, we will introduce the necessary definitions and some basic results related to them but let us first recall the definition of a metric.
Definition 2.1. For any non-empty space G, a metric is a function d : G × G → [0, ∞) that fulfills the following three conditions for all x, y, z ∈ G: (1) Positivity: d(x, y) ≥ 0, and d(x, y) = 0 if and only if x = y, A quasi-metric is a function d that fulfills the definition above otherwise, but instead of the triangle inequality itself, it only fulfills the inequality with some constant c > 1.
Now, let us introduce the notations used. Suppose that G R n is some domain. For all x ∈ G, the Euclidean distance d(x, ∂G) = inf{|x − z| | z ∈ ∂G} will be denoted by d G (x). The Euclidean balls and spheres are written as B n (x, r) = {y ∈ R n | |x − y| < r}, B n (x, r) = {y ∈ R n | |x − y| ≤ r} and S n−1 (x, r) = {y ∈ R n | |x − y| = r}. If x or r is not specified otherwise, suppose that x = 0 and r = 1. Furthermore, the Euclidean line passing through points x, y ∈ R n is denoted by L(x, y) and the unit vectors by {e 1 , ..., e n }.
In this paper, we focus on the cases where the domain G is either the upper halfplane H n = {(x 1 , ..., x n ) ∈ R n | x n > 0}, the unit ball B n = B n (0, 1) or the open sector S θ = {x ∈ C | 0 < arg(x) < θ} with an angle θ ∈ (0, 2π). The hyperbolic metric can be defined in these cases with the formulas , x, y ∈ B n , ρ S θ (x, y) = ρ H 2 (x π/θ , y π/θ ), x, y ∈ S θ , see [5, (4.8), p. 52 & (4.14), p. 55]. In the two-dimensional unit disk, we can simply write where y is the complex conjugate of y. For any domain G R n , define the following hyperbolic type metrics and quasi-metric:   In particular, when defined in a sector S θ , they are invariant under a reflection over the bisector of the sector and a stretching x → r · x with any r > 0. Consequently, this allows us to make certain assumptions when choosing the points x, y ∈ S θ .
The metrics introduced above fulfill the following inequalities.

Point pair function
In this section, we will focus on the point pair function. The expression for this function was first introduced in [2, p. 685], but it was named and researched further in [6]. It was noted already in [2, Rmk 3.1 p. 689] that the point pair function defined in the unit disk is not a metric because it does not fulfill the triangle inequality for all points of this domain. However, the point pair function offers a good upper bound for the triangular ratio metric in convex domains [5, Lemma 11.6(1), p. 197] and, by Lemma 2.5, it also serves as a lower bound for the expression th(ρ G (x, y)/2) if G ∈ {H n , B n } so studying its properties more carefully is relevant.
It is very easy to show that there is a constant c > 1 with which the point pair is a quasi-metric. Proof. It follows from Lemma 2.4(1) and the fact that the j * G -metric is always a metric that However, even for an arbitrary domain G R n , the constant √ 2 is not the best possible. This is because, as we show in Theorem 3.7, this number can be replaced by a smaller constant √ 5/2. Now, let us introduce two quite simple results in order to prove Theorem 3.7.
Proposition 3.2. If the points x, z ⊂ R n are fixed and x ′ ∈ S n−1 (x, r) for some constant and the equality here holds if and only if |z − x ′ | = r + |x − z|.
Proof. Since the distances |x − z| and |x − x ′ | = r are fixed, the minimum value of the first quotient above is obtained by maximising the distance |z − x ′ |. By the triangle inequality, |z − x ′ | ≤ |x − z| + |x − x ′ | and the equality here holds if and only if x ′ ∈ L(x, z) such that x ∈ [z, x ′ ]. Thus, the result follows.
R and, in this case, the supremum is obtained when Proof. Let us first consider the question without specifying the domain G R n or its boundary in any way. Instead, fix distinct points x, y ∈ R n and choose distances d 0 , is fixed.
Next, choose z ∈ R n and suppose that there are some points x ′ ∈ S n−1 (x, d 0 ) and y ′ ∈ S n−1 (y, d 1 ) but do not fix these points x ′ , y ′ otherwise. Just like in Proposition 3.2, the minimum value of the sum Since this sum is increasing with respect to |x − z| and |z − y|, we need to choose z ∈ [x, y] to minimize it and, thus, the line L(x, y) must contain the points x ′ , x, z, y, y ′ in this order. Finally, consider some domain G R n . For all x, y ∈ G, there are some points The expression for p G (x, y) is equivalent to the quotient (3.5), and we need to find the minimum value for the sum p G (x, z) + p G (z, y). From the expression of the function p G , we see that this sum is is clearly decreasing with respect to d G (z). There only needs to be points (3.6), and it follows from above that it is minimized when x ′ , x, z, y, y ′ ∈ L(x, y) in this order. Furthermore, we can now set n = 1 without loss of generality and the result follows.
With the lemma above, we can now prove the best constant with which the point pair function is a quasi-metric, assuming that the domain is not fixed. Proof. We are interested in the greatest value of the quotient (3.4) so, by Lemma 3.3, we we will have by the equality (3.8) that and, further, the value of the sum p G (x, z) + p G (x, y) can be described with the function Denote now , Here, the function j(h) is the value of the distance We will have Consider the quotient (3.9). Increasing the value of h lessens quite similarly the numerator and the second term in the denominator. However, the first term of the denominator is clearly larger than the second term and it does not depend on h. Thus, we can see that the quotient (3.9) is decreasing with respect to h. With a similar argument, we can also show that the quotient (3.10) is increasing with respect to h.
From these observations, it follows that .
Above, the last equality follows from the fact that l 0 (k, 2k − 1) = l 1 (k, 2k − 1) for all 1/2 ≤ k ≤ 1. This is because h = 2k − 1 is equivalent to k = (h + 1)/2 and the function l(k, h) is continuous here. Next, we need to calculate the supremums above and find out which one of them is greater. Let us first find the minimum value of the function l 0 (k, 0) for 0 ≤ k ≤ 1/2. From the definition of l 0 (k, h), we will have .
Similarly, by differentiation and the quadratic formula, the only stationary point on the interval k ∈ (1/2, 1) is Thus, the greatest value of the quotient q 1 (k) with respect to k is either By substituting the quotients q 0 (k) and q 1 (k) in (3.13) with their greatest values found above, we will have and, by combining this with (3.11) and (3.12), sup 0≤h≤1, 0≤k≤1 and it can be shown that, for all d G (x) ≤ 0, Consequently, which means that p G is a quasi-metric with a constant at most √ 5/2. Let us yet show that this constant is sharp. Suppose now that we have a domain G = (−1, 1) R 1 . If x = −1/3, y = 1/3 and z = 0, we will have so the result follows.
Let us now show exactly when the constant of Theorem 3.7 is the best possible, assuming the domain G is fixed.
Proposition 3.14. If there are some k ∈ G R n and r > 0 such that B n (k, r) ⊂ G and d(S n−1 (k, r) ∩ ∂G) = 2r, then √ 5/2 is the best constant with which p G is a quasi-metric.
An example of a domain where the constant √ 5/2 is truly the best possible is the unit ball B n . We can easily see this by choosing x = e 1 /3, z = 0 and y = −e 1 /3. Other domains like this include, for instance, a twice punctured space R n \({s}∪{t}), s = t ∈ R n , and all k-dimensional hypercubes in R n where 1 ≤ k ≤ n. However, note that there are also domains in which √ 5/2 is not the best possible: For instance, by Lemma 2.5(1), p H n (x, y) = s H n (x, y), so the point pair function is a metric in a domain G = H n . It also follows from this that the point pair function is a metric in a sector with an angle over π. Theorem 3.15. In an open sector S θ with an angle π ≤ θ < 2π, the point pair function p S θ is a metric.
Proof. Trivially, we only need to prove that the point pair function fulfills the triangle inequality in this domain. Fix distinct points x, y ∈ S θ . Note that if θ ≥ π then, for every point x ∈ S θ , there is exactly one point x ′ ∈ S 1 (x, d S θ (x)) ∩ ∂S θ . Fix x ′ , y ′ like this for the points x, y, respectively. Furthermore, denote the closed hull The special case where where x, y, x ′ , y ′ are collinear and J is just a line is possible, but this does not affect our proof.
We are interested in such a point z ∈ S θ that minimizes the sum p S θ (x, z) + p S θ (z, y). This point z must belong in J ∩ S θ : If z / ∈ J, then it can be rotated around either x or y into a new point z ∈ J ∩ S θ so that one of the distances |x − z| and |z − y| does not change and the other one decreases, and the distance d S θ (z) increases.
Choose now a half-plane H such that x, y ∈ H and ∂H is a tangent for both S 1 (x, d S θ (x)) and S 1 (y, d S θ (y)). Now, d H (x) = d S θ (x) and d H (y) = d S θ (y). Clearly, J ∩ S θ ⊂ H and, since θ ≥ π, for every point z ∈ J ∩S θ , d S θ (z) ≤ d H (z). Recall that the point pair function p G is a metric in a half-plane domain. It follows that which proves our result.
It can be shown that the point pair function p G is not a metric in a sector S θ with an angle 0 < θ < π. For instance, if θ = π/2, then the points x = e π/5 , y = e 3π/10 and z = (x + y)/2 do not fulfill the triangle inequality. However, it is still useful to study this function in sector domains because it has several beneficial properties that can be used when creating bounds for hyperbolic type metrics. Next, we will show that the point pair function is invariant under a certain conformal mapping defined in a sector domain. Note that the triangular ratio metric has this same property, see [12,Thm 4.16, p. 14]. Proof. By Remark 2.3, we can fix x = e ki and y = re hi with r > 0 and 0 < k ≤ h < θ without loss of generality. Now, f (x) = x = e ki and f (y) = (1/r)e hi . It follows that which proves the result.
Let us yet consider the connection between the point pair function and the triangular ratio metric and, especially, what we can tell about the domain by studying these metrics. Theorem 3.18. If G R n is a domain such that s G (x, y) = p G (x, y) holds for all x, y ∈ G, then the following conditions are fulfilled: Proof. (1) If the equality s G (x, y) = p G (x, y) holds for all x, y ∈ G, it follows from Theorem 3.17 that G is convex.
with some very small distance r > 0. It follows from the first part of this theorem that the convex hull ∪ u∈B n (x,r), v∈B n (y,r) [u, v] must belong to G. Suppose that each point q inside this bounded area fulfills d(q, [u ′ , v ′ ]) ≤ d(q, ∂G); if not, choose a smaller r.
Consider now the quotient .
Thus, the quotient (3.19) is truly smaller than 1 and the equality s G (x, y) = p G (x, y) cannot hold.
The next result follows. Corollary 3.20. If G R n is a domain such that s G (x, y) = p G (x, y) holds for all x, y ∈ G, then G can be transformed into a half-space with only translations and rotations.

New quasi-metric
In this section, we define a new intrinsic quasi-metric w G in a convex domain G and study its basic properties. As can be seen from Theorem 4.5, this function gives a lower bound for the triangular ratio metric. Since the point pair function serves as an upper bound for the triangular ratio metric, these two quasi-metrics can be used to form bounds for the triangular ratio distance like in Corollary 4.7. Furthermore, these three functions are equivalent in the case of the half-space, see Proposition 4.2, so these bounds are clearly essentially sharp at least in some cases.
First, consider the following definition.
Definition 4.1. Let G R n be a convex domain. For any x ∈ G, there is a non-empty set Define now a function w G : Note that we can only define the function w G for convex domains G because, for a non-convex domain G and some points x, y ∈ G, there is some points x, y ∈ G such that y = x with some x ∈ X and the denominator in the expression of w G would become zero. Proof. For all x = (x 1 , ..., x n ) ∈ H n , there is only one point x = (x 1 , ..., x n−1 , −x n ) = x − 2x n in the set X. Thus, The result s H n (x, y) = p H n (x, y) is in Lemma 2.5(1).
While it trivially follows from the result above that the function w G is a metric in the case G = H n , this is not true for all convex domains G, as can be seen with the following example.
Proof. Let x = 1/2 + k + i/2, y = −1/2 + i/2 and z = −1/2 − k + i/2 with 0 < k < 1/3. Now, It follows also from Example 4.3 that there cannot be a constant less than √ 2 with which the function w G would be a quasi-metric for all domains. In fact, we will prove in Corollary 4.6 that, for an arbitrary convex domain, w G is a quasi-metric with the constant √ 2. However, in order to do this, let us first consider two inequalities.
Proposition 4.4. For any convex domain G R n and all x, y ∈ G, j * G (x, y) ≤ w G (x, y). Proof. By the triangle inequality, so the result follows.
Proof. Choose any distinct x, y ∈ G. By symmetry, we can suppose that inf x∈ X |y − x| ≤ inf y∈ Y |x − y|. Fix x as the point that gives this smaller infimum. Let us only consider the two-dimensional plane containing x, y, x and set n = 2. Fix u = [x, x] ∩ ∂G and z = [y, x] ∩ ∂G. Position the domain G on the upper half-plane so that the real axis is the tangent of S 1 (x, d G (x)) at the point u. Since G is convex, it must be a subset of H 2 and therefore z ∈ H 2 ∪ R. Thus, it follows that |x − z| ≤ |z − x|. Consequently, y).
The inequality s G (x, y) ≤ √ 2w G (x, y) follows from Lemma 2.4(3) and Proposition 4.4. Now, we can show that the function w G is a quasi-metric.
Corollary 4.6. For an arbitrary convex domain G R n , the function w G is a quasimetric with a constant greater than or equal to √ 2, and this is constant is the best possible.
Proof. It follows from Theorem 4.5 and the fact that the triangular ratio metric is always a metric that and, by Example 4.3, the constant √ 2 here is the best for an arbitrary convex domain.
We will also have the following result.  y). Proof. Follows from Proposition 4.4 and Theorems 4.5 and 3.17.

Quasi-metrics in the unit disk
In this section, we will focus on the inequalities between the hyperbolic type metrics and quasi-metrics in the case of the unit disk. Calculating the exact value of the triangular ratio metric in the unit disk is not a trivial task, but instead quite a difficult problem with a very long history, see [3] for more details. However, we already know from Corollary 4.7 that the quasi-metric w G serves as a lower bound for the triangular ratio metric in convex domains G and this helps us considerably.
Remark 5.1. Note that while we focus below just on the unit disk B 2 , all the inequalities can be extended to the general case with the unit ball B n , because the values of the metrics and quasi-metrics considered only depend on how the points x, y are located on the twodimensional place containing them and the origin.
First, we will define the function w G of Definition 4.1 in the case G = B 2 . Denote below x = x(2 − |x|)/|x| for all points x ∈ B 2 \{0}. We will have the following results.
Proof. By writing µ = ∡XOY and using the law of cosines, Now, consider the following definition.
By Proposition 5.2, for all points x, y ∈ B 2 such that 0 < |y| ≤ |x| < 1, It follows from this and Proposition 5.3 that the function w B 2 of Definition 5.4 is continuous. It is also easy to verify that this function is truly equivalent to that of Definition 4.1 and, by Corollary 4.6, the function w B 2 truly is at least a quasi-metric. In fact, according to the numerical tests, the function w B 2 seems to fulfill the triangle inequality on the unit disk, which would mean that the following conjecture holds.
Conjecture 5.5. The function w B 2 is a metric on the unit disk.
However, it does not affect the results of this paper if the function w B 2 truly is a metric or just a quasi-metric, since we use it to create new inequalities between it and some hyperbolic type metrics. Thus, let us move on and show instead that this function w B 2 is quite a good lower bound for the triangular ratio metric in the unit disk. First, before proving Theorem 5.7, note that the value of the triangular ratio metric can be calculated for the collinear points with the help of the following lemma.
where the equality holds if the points x, y are collinear with the origin.
Theorem 5.7. For all x, y ∈ B 2 , w B 2 (x, y) ≤ s B 2 (x, y) and the equality holds here whenever x, y are collinear with the origin.
Proof. The inequality follows from Theorem 4.5. Suppose now that the points x, y ∈ B 2 \{0} are collinear with the origin. Without loss of generality, we can fix 0 < y < 1, so that |x − y| = |x − (2 − y)|. Now, by Lemma 5.6, By combining this inequality (5.8) with the one in Theorem 4.5, we will have the equality s B 2 (x, y) = w B 2 (x, y) for all points x, y ∈ B 2 \{0} collinear with the origin. In the special case where y = 0, so the theorem follows.
By [13,Lemma 3.12,p. 7], the following function can be used to create a lower bound for the triangular ratio metric. However, the quasi-metric w B 2 is a better lower bound for the triangular ratio metric in the unit disk than this low-function. Lemma 5.10. For all distinct points x, y ∈ B 2 \{0}, w B 2 (x, y) > low(x, y).
Proof. Let x ∈ B 2 , k > 1 and µ = ∡([x, 1], [1, k]). Now, Since π/2 < µ ≤ π, cos(µ) < 0. Thus, we see that the distance |x−k| is strictly increasing with respect to k. In other words, the further away a point k ∈ R 2 \B 2 is from the origin, the longer the distance between k and an arbitrary point x ∈ B 2 is. For every point y ∈ B 2 \{0}, so it follows by the observation made above that, for all x ∈ B 2 , |x − y| < |x − y * |.
Consequently, by symmetry, the inequality holds for all points x, y ∈ B 2 \{0}.
Next, we will prove one sharp inequality between the two quasi-metrics considered this paper.
Proof. For all points x, y ∈ B 2 , the inequality w B 2 (x, y) ≤ p B 2 (x, y) follows from Corollary 4.7. Let us now prove that the equality in the case where x, y are on the same radius. If 0 < y ≤ x < 1, and, in the special case y = 0, Next, let us prove the latter part of the inequality. We need to find that the maximum of the quotient . (5.13) In order to do that, we can suppose without loss of generality that x, y are on different radii since, as we proved above, the equality w B 2 (x, y) = p B 2 (x, y) holds otherwise. Choose these points so that 0 < |y| ≤ |x| < 1 and µ = ∡XOY . It follows from Proposition 5.2 that the quotient (5.13) is now . (5.14) Fix now j = cos(µ), s = |y| 2 + (2 − |x|) 2 , t = 2|y|(2 − |x|), so that the quotient that is the argument of the square root in the expression (5.14) can be described with a function f : [0, 1] → R, By differentiation, the function f is decreasing with respect to j, if Since this last inequality is equivalent to which clearly holds, it follows that the function f and the quotient (5.13) are decreasing with respect to j = cos(µ). The minimum value of cos(µ) is −1 at µ = π. Consequently, we can fix the points x, y so that x = h and y = −h + k with 0 ≤ k < h < 1, without loss of generality. Now, the quotient (5.13) is This upper bound found above is the value of the quotient (5.13) in the case k = 0, because, for x = h and y = −h, The expression 2h 2 − 2h + 1 obtains its minimum value 1/4 at h = 1/2, so the maximum value of the quotient (5.13) is √ 2.
The next result follows.  , y), where the equality j * B 2 (x, y) = w B 2 (x, y) holds whenever x, y are on the same radius and the inequality w B 2 (x, y) ≤ √ 2j * B 2 (x, y) has the best constant possible. Proof. The inequality j * B 2 (x, y) ≤ w B 2 (x, y) follows from Proposition 4.4 and the inequality w B 2 (x, y) ≤ √ 2j * B 2 (x, y) from Lemma 2.4(1) and Theorem 5.11. Consider now the case where x, y are on the same radius, but neither of them is the origin. Without loss of generality, we can suppose that 0 < y ≤ x < 1 and now If y = 0 instead, then w B 2 (x, 0) = |x| 2 − |x| = |x| |x| + 2(1 − |x|) = j * B 2 (x, y).
Next, let us show that the constant √ 2 in the theorem is the best possible. Fix x = 1 − k and y = (1 − k)e ki for some 0 < k < 1. Now, Let us now focus on how the quasi-metric w B 2 can be used to create an upper bound for the triangular ratio metric. We know from Theorem 4.5 that in the general case where the domain G is convex, the inequality s G (x, y) ≤ √ 2w G (x, y) holds. Thus, this must also hold in the unit disk, but several numerical tests suggest that the constant √ 2 is not necessarily the best possible when G = B 2 . The next result tells the best constant in a certain special case. Lemma 5.16. For all x, y ∈ B 2 such that |x| = |y| and ∡XOY = π/2, s B 2 (x, y) ≤ c · w B 2 (x, y) with c = h 2 0 − 2h 0 + 2 Proof. Let x = h and y = hi for 0 < h < 1. Now, Then By differentiation, By the quadratic formula, f ′ (h) = 0 holds when