Introducing a new intrinsic metric

A new intrinsic metric called $t$-metric is introduced. Several sharp inequalities between this metric and the most common hyperbolic type metrics are proven for various domains $G\subsetneq\mathbb{R}^n$. The behaviour of the new metric is also studied under a few examples of conformal and quasiconformal mappings, and the differences between the balls drawn with all the metrics considered are compared by both graphical and analytical means.


Introduction
In geometric function theory, one of the topics studied deals with the variation of geometric entities such as distances, ratios of distances, local geometry and measures of sets under different mappings. For such studies, we need an appropriate notion of distance that is compatible with the class of mappings studied. In classical function theory of the complex plane, one of the key concepts is the hyperbolic distance, which measures not only how close the points are to each other but also how they are located inside the domain with respect to its boundary.
The hyperbolic distance also serves as a model when we need generalisations to subdomains G of arbitrary metric spaces X. These generalized distances behave like the hyperbolic metric in the aspect that they define the Euclidean topology and, in particular, we can cover compact subsets of G using balls of the generalized metrics. Thus, the boundary of the domain has a strong influence on the inner geometry of the domain defined by some chosen metric.
Since the classical hyperbolic geometry acts as a model, some of its key features are inherited by the generalizations but not all. For instance, it is desirable to study local behaviour of functions and we need to have a metric that is locally comparable with the Euclidean geometry. Such a metric is here called an intrinsic metric. Note that there is no established definition for this concept and it is sometimes required, for instance, that the closures of the balls defined with an intrinsic metric never intersect the boundary of the domain. An example of an intrinsic metric is the following new metric, on which this work focuses.
File: main.tex, printed: 2021-12-3, 18.46 Definition 1.1. Let G be some non-empty, open, proper and connected subset of a metric space X. Choose some metric η G defined in the closure of G and denote η G (x) = η G (x, ∂G) = inf{η G (x, z) | z ∈ ∂G} for all x ∈ G. The t-metric for a metric η G in a domain G is a function t G : G × G → [0, 1], for all x, y ∈ G. Here, we mostly focus on the special case where G R n and η G is the Euclidean distance.
Our work in this paper is motivated by the research of several other mathematicians. During the past thirty years, many intrinsic metrics have been introduced and studied [3,4,7,10,11]. It is noteworthy that each metric might be used to discover some intricate features of mappings not detected by other metrics. Since our new metric differs slightly from other intrinsic metrics and has a relatively simple definition, it could potentially be a great help for new discoveries about intrinsic geometry of domains. For instance, there is one inequality that is an open question for the triangular ratio metric, but could potentially be proved for the t-metric, see Conjecture 4.13 and Remark 4.14.
Unlike several other hyperbolic type metrics, such as the triangular ratio metric or the hyperbolic metric itself, the t-metric does not have the property about the closed balls never intersecting with the boundary, see Theorem 5.5. This is an interesting aspect for this metric clearly fulfills most of the others, if not all, properties of a hyperbolic type metrics listed in [7, p. 79]. Consequently, we have found an intrinsic metric that does not have one of the common properties of hyperbolic type metrics.
In this paper, we will study this new metric and its connection to other metrics. In Section 3, we prove that the function of Definition 1.1 is really a metric and find the sharp inequalities between this metric and several hyperbolic type metrics, including also the hyperbolic metric, in different domains. In Section 4, we show how the t-metric behaves under certain quasiconformal mappings and find the Lipschitz constants for Möbius maps between balls and half-spaces. Finally, in Section 5, we draw t-metric disks and compare their certain properties to those of other metric disks.
Acknowledgements. The research of the first author was supported by Finnish Concordia Fund.

Preliminaries
In this section, we will introduce the definitions of a few different metrics and metric balls that will be necessary later on but, first, let us recall the definition of a metric. Definition 2.1. For any non-empty space G, a metric is a function η G : G × G → [0, ∞) that fulfills the following three conditions for all x, y, z ∈ G: (1) Positivity: η G (x, y) ≥ 0, and η G (x, y) = 0 if and only if x = y, (2) Symmetry: η G (x, y) = η G (y, x), Let η G be now some arbitrary metric. An open ball defined with it is B η (x, r) = {y ∈ G | η G (x, y) < r} and the corresponding closed ball is B η (x, r) = {y ∈ G | η G (x, y) ≤ r}. Denote the sphere of these balls by S η (x, r). For Euclidean metric, these notations are B n (x, r), B n (x, r) and S n−1 (x, r), respectively, where n is the dimension. In this paper, the unit ball B n = B n (0, 1), the upper half-plane H n = {(x 1 , ..., x n ) ∈ R n | x n > 0} and the open sector S θ = {x ∈ C | 0 < arg(x) < θ} with an angle θ ∈ (0, 2π) will be commonly used as domains G. Note also that the unit vectors will be denoted by {e 1 , ..., e n }.
Let us now define the metrics needed for a domain G R n . Denote the Euclidean distance between the points x, y by |x − y| and let d G (x) = inf{|x − z| | z ∈ ∂G}. Suppose that the t-metric is defined with the Euclidean distance so that for all x, y ∈ G, if not otherwise specified.
The following hyperbolic type metrics will be considered: The triangular ratio metric: and the point pair function: Out of these hyperbolic type metrics, the triangular ratio metric was studied by P. Hästö in 2002 [9], and the two other metrics are more recent. As pointed out in [8], the j * G -metric is derived from the distance ratio metric found by F.W. Gehring and B.G. Osgood in [6]. Note that there are proper domains G in which the point pair function is not a metric [3, Rmk 3.1 p. 689].
Define also the hyperbolic metric as , x, y ∈ H n , , x, y ∈ B n in the upper half-plane H n and in the Poincaré unit disk B n [7, (4.8), p. 52 & (4.14), p. 55]. In the two-dimensional space,

t-Metric and Its Bounds
Now, we will prove that our new metric is truly a metric in the general case.
Theorem 3.1. For any metric space X, a domain G X and a metric η G defined in G, the function t G is a metric.
Proof. The function t G is a metric if it fulfills all the three conditions of Definition 2.1. Trivially, the first two conditions hold. Consider now a function f : for all x, y, z ∈ G. Thus, t G fulfills the triangle inequality.
We now show that the method of proof of Theorem 3.1 can be used to prove that several other functions are metrics, too.
for all x, y ∈ G, is a metric in the domain G.
Proof. Since η G is a metric and c G is both symmetric and non-negative, the function φ G trivially fulfills the first two conditions of Definition 2.1. Note that, by the triangle inequality of the metric η G and the inequality (3.3), the inequalities so the function φ G fulfills the triangle inequality and it must be a metric.

Remark 3.4.
(1) If the function c G of Theorem 3.2 is strictly positive, the condition φ G (x, x) = 0 does not need to be separately specified. Namely, this condition follows directly from the fact that η G (x, x) = 0 for a metric η G . Note also that if c G is a null function, the function φ G becomes the discrete metric.
for all x, y ∈ B n with a constant 0 < c ≤ 1, is a metric on the unit ball.
Proof. Since now for all x, y, z ∈ B n , the result follows from Theorem 3.2.
so the inequality (3.3) holds for all x, y, z ∈ G and the result follows from Theorem 3.2.
for all x, y ∈ B n with a constant 0 < c ≤ √ 2, is a metric on the unit ball.
Let us focus again on the t-metric. Since the result of Theorem 3.1 holds for any metric η G , the t-metric is trivially a metric also when defined for the Euclidean metric. Below, we will consider the t-metric in this special case only. Let us next prove the inequalities between the t-metric and the three hyperbolic type metrics defined earlier.
Theorem 3.8. For all domains G R n and all points x, y ∈ G, the following inequalities hold: Furthermore, in each case the constants are sharp for some domain G.
With this information, we can write The equality t G (x, y) = j * G (x, y) holds always when d G (x) = d G (y). For x = i and y = ki, lim k→0 + (t H 2 (x, y)/j * H 2 (x, y)) = 1/2. Thus, the first inequality of the theorem and its sharpness follow.
Next, we will study the connection between the t-metric and the hyperbolic metric.
Theorem 3.11. For all x, y ∈ G ∈ {H n , B n }, the inequality Proof. In the case G = H n , the result follows directly from Lemma 2.4(1) and Theorem 3.8 (3).
Since the values of the t-metrics and the hyperbolic metric in the domain B n only depend on how the points x, y are located on the two-dimensional plane fixed by these two points and the origin, we can assume without loss of generality that n = 2. Consider now the quotient which proves that th(ρ B 2 (x, y)/2)/2 ≤ t B 2 (x, y). By the observation above, this also holds in the more general case where n is not fixed. The inequality is sharp, too: For x = ke 1 and y = −ke 1 , lim k→0 + (t B n (x, y)/th(ρ B n (x, y)/2)) = 1/2.

Quasiconformal Mappings and Lipschitz Constants
In this section, we will study the behaviour of the t-metric under different conformal and quasiconformal mappings in order to demonstrate how this metric works.
Remark 4.1. The t-metric is invariant under all similarity maps. In particular, the tmetric defined in a sector S θ is 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 .
First, let us study how the t-metric behaves under a certain conformal mapping between two sectors with angles at most π.

(4.5)
The value of the quotient above is at greatest, when k − h is at minimum and both γ and µ are at maximum. This happens when h < α/2 and k = α − h. Now, γ = µ = h and the quotient (4.5) is .
Since the expression above is strictly increasing with respect to h and h < α/2, the maximum value of the quotient (4.4) is which, together with the inequality (4.3), proves the first part of our theorem. Suppose next that α > β instead. It can be now proved that the minimum value of the quotient (4.4) is the same limit value (4.6) and, by Theorem 3.8(3) and [12, Lemma 5.11, p. 13], t S β (f (x), f (y)) ≤ 2t Sα (x, y). Thus, the theorem follows.
Let us now consider a more general result than the one above. Namely, instead of studying a conformal power mapping, we can assume that, for domains G 1 , G 2 ⊂ R 2 , the mapping f : G 1 → G 2 = f (G 1 ) is a K-quasiconformal homeomorphism, see [13,Ch. 2]. Let c(K) be as in [7,Thm 16.39,p. 313]. Now, c(K) ≥ K and c(K) → 1 whenever K → 1. See also the book [5] by F.W. Gehring and K. Hag.
Next, we will focus on the radial mapping, which is another example of a quasiconformal mapping, see [13, 16.2, p. 49].
Theorem 4.8. If f : G → G with G = B 2 \{0} is the radial mapping defined as f (z) = |z| a−1 z for some 0 < a < 1, then for all x, y ∈ G such that |x| = |y|, the sharp inequality holds.
If 0 < r ≤ 1/2 < r a < 1, the quotient (4.9) is , (4.10) which is decreasing with respect to k. Since lim k→0 + r a (r sin(k) + r) r(r a sin(k) + 1 − r a ) = r 1+a r(1 − r a ) = r a 1 − r a is increasing with respect to r and r ≤ 1/2, the maximum value of the quotient (4.10) is 1/(2 a − 1). The other limit value lim k→1 − r a (r sin(k) + r) r(r a sin(k) + 1 − r a ) = r a (r + r) r(r a + 1 − r a ) = 2r a is increasing with respect to r a and r a > 1/2, so the quotient (4.10) is always more than 1. If 1/2 < r < r a < 1, the quotient (4.9) is = r a (r sin(k) + 1 − r) r(r a sin(k) + 1 − r a ) , (4.11) which is decreasing with respect to k. Since r > 1/2 and lim k→0 + r a (r sin(k) + 1 − r) r(r a sin(k) is decreasing with respect to r, the quotient (4.10) is less than 1/(2 a − 1). The other limit value is lim k→1 − r a (r sin(k) + 1 − r) r(r a sin(k) + 1 − r a ) = r a−1 , which is clearly more than 1. Thus, the minimum value of the quotient (4.9) is 1 and the maximum value 1/(2 a − 1), so the theorem follows.
Let us now find Lipschitz constants of a few different mappings for the t-metric.
holds for all x, y ∈ G 1 .
Proof. By Theorem 3.11 and the conformal invariance of the hyperbolic metric, It follows from Theorem 4.12 that the Lipschitz constant Lip(f |G 1 ) for the t-metric in any conformal mapping f : , h(y)) t B n (x, y) = lim k→1 − (k + 1) = 2, the Lipschitz constant Lip(h|B 2 ) is equal to 2. However, for certain Möbius transformations, there might be a better constant than 2. For instance, the following conjecture is supported by several numerical tests.  In the next few results, we will study a mapping f * : S θ → S θ , f * (x) = x/|x| 2 defined in some open sector S θ , and find its Lipschitz constants for the t-metric. Theorem 4.15. If θ ∈ (0, π] and f * is the mapping f * : S θ → S θ , f * (x) = x/|x| 2 , the Lipschitz constant Lip(f * |S θ ) for the t-metric is 1 + sin(θ/2).
Proof. Without loss of generality, we can fix x = e hi and y = re ki with 0 < h ≤ π/2, h ≤ k < θ and r > 0. Since x * = e hi and y * = (1/r)e ki , it follows that To maximize this, we clearly need to choose k = θ/2 and make r and h as small as possible. If k = θ/2, so the theorem follows.
Proof. It follows from Theorems 4.16 and 3.8(3) that for all x, y ∈ S θ . Since for x = e hi and y = re πi/2 with h < π/2 and r > 0, and it follows that

Comparison of Metric Balls
Next, we will graphically demonstrate the differences and similarities between the various metrics considered in this paper by drawing for each metric several circles centered at the same point but with different radii. In all of the figures of this section, the domain G ⊂ R 2 is a regular five-pointed star and the circles have a radius of r = 1/10, ..., 9/10. The center of these circles is in the center of G in the first figures, and then off the center in the rest of the figures. All the figures in this section were drawn by using the contour plot function contour in R-Studio and choosing a grid of the size 1,000×1,000 test points. While we graphically only inspect circles and disks, we will also prove some properties for the n-dimensional metric balls.
For several hyperbolic type metrics, the metric balls of small radii resemble Euclidean balls, but the geometric structure of the boundary of the domain begins to affect the shape of these balls when their radii grow large enough, see [7,Ch. 13,. By analysing this phenomenon more carefully, we can observe, for instance, that the balls are convex with radii less than some fixed r 0 > 0 in the case of some other metrics, see [7,Thm 13.6,p. 241; Thm 13.41 p. 256; Thm 13.44, p. 258]. From the figures of this section, we see that the four metrics studied here share this same property. In particular, we notice that, while the metric disks with small radii are convex and round like Euclidean disks, the metric circles with larger radii are non-convex and have corner points. By a corner point, we mean here such a point on the circle arc that has many possible tangents.
In the following theorem, we will prove a property that can be seen from Figures 1c, 1d, 2c and 2d.
Theorem 5.1. If the domain G is a polygon, then the corner points of the circles S p (x, r) and S t (x, r) are located on the the angle bisectors of G.
Proof. Suppose G has sides l 0 and l 1 that have a common endpoint k. Fix x ∈ G and choose some point y ∈ G so that k is the vertex of G that is closest to y and there is no other side closer to y than l 0 and l 1 . Thus, d G (y) = min{d(y, l 0 ), d(y, l 1 )} and, for a fixed distance |x − y|, d G (y) is at maximum when d(y, l 0 ) = d(y, l 1 ). The condition d(y, l 0 ) = d(y, l 1 ) is clearly fulfilled when y is on the bisector of ∠(l 0 , l 1 ) and, the greater the d G (y), the smaller the distances p G (x, y) and t G (x, y) are now. Consequently, if the (a) s G -metric circles.
(c) p G -metric circles.  circle S p (x, r) or S t (x, r) has a corner point, it must be located on an angle bisector of G.
However, it can been seen from Figures 2a and 2b that the circles with s G -and j * Gmetrics can have corner points also elsewhere than on the angle bisectors of the domain (a) s G -metric circles.
(c) p G -metric circles.  G. We also notice that the circles in Figure 2b clearly differ those in Figures 2a and 2c. This can be described with the concept of starlikeness, which is a looser form of convexity. Namely, a set K is starlike with respect to a point x ∈ K if and only if the segment [x, y] belongs to K fully for every y ∈ K. In particular, the five-pointed star domain is starlike with respect to its center. The disks by the j * G -and t G -metrics (Figures 2a and 2c) are clearly not starlike and, even if it cannot be clearly seen from Figure 2d, there are disks drawn with the point pair function p G that are not starlike.
Lemma 5.2. There exist disks B j * (x, r), B p (x, r), and B t (x, r) that are not starlike with respect to their center.
The segment [x, y] does not clearly belong to G fully and no disk in G can contain this segment. However, its end point y is clearly included in the disks B j * (x, 0.7), B t (x, 0.7) and B p (x, 0.91). Thus, we have found examples of non-starlike disks.
There are no disks or balls like this for the triangular ratio metric. For several common hyperbolic type metrics η G , the closed ball B η (x, M ) with M = η G (x, y) and x, y ∈ G is always a compact subset of the domain G, see [7, p. 79]. For instance, the hyperbolic metric ρ G has this property [7, p. 192]. As can be seen from the figures, the j * -metric, the triangular ratio metric and the point pair function share this property, too. Proof. If the ball B η (x, r), η G ∈ {j * G , p G , s G }, touches the boundary of G, then there is some point y ∈ S η (x, r) with d G (y) = 0 and j * G (x, y) = p G (x, y) = s G (x, y) = 1. Thus, we need to just prove that the balls with radius 1 always touch the boundary. Consider first the balls B j * (x, r) and B p (x, r), with a radius r = 1. Since, for all the points y on their boundary, j * G (x, y) = 1 ⇔ 2 min{d G (x), d G (y)} = 0 ⇔ d G (x) = 0 or d G (y) = 0, p G (x, y) = 1 ⇔ 4d G (x)d G (y) = 0 ⇔ d G (x) = 0 or d G (y) = 0, the balls B j * (x, 1) and B p (x, 1) touch the boundary of G.
Consider yet the triangular ratio metric. Because only balls with radius r = 1 can touch the boundary, B s (x, 1) ∩ ∂G = ∅. However, if s G (x, y) = 1, there is some point z ∈ ∂G such that |x − y| = |x − z| + |z − y|. This means that z is on a line segment [x, y] and, since z / ∈ B s (x, 1), z must be arbitrarily close to the point y. Thus, d G (y) = 0 and the ball B s (x, 1) touches the boundary.
However, the t-metric differs from the hyperbolic type metrics in this aspect: the closure of a t-metric ball is a compact set, if and only if the radius of the ball is less than 1/2. Theorem 5.5. The balls B t (x, r) touch the boundary of the domain G R n if and only if r ≥ 1 2 . Proof. If B t (x, r) touches the boundary, there must be some y ∈ S t (x, r) such that d G (y) = 0. Since d G (x) ≤ |x − y| + d G (y), it follows that Thus, only balls B t (x, r) with a radius r ≥ 1 2 can touch the boundary of G. Let us yet prove that the balls B t (x, 1 2 ) always touch the boundary of G. For any point y ∈ S t (x, 1 2 ), it holds that |x − y| = 1/2 and Since only balls B t (x, r) with r ≥ 1 2 can touch the boundary of G, B t (x, 1 2 ) ∩ ∂G = ∅ and d G (x) ≥ 1/2. Thus, d G (y) = 1/2 − d G (x) ≤ 0 and, since the distances cannot be negative, d G (y) = 0 and the ball B t (x, 1 2 ) truly touches the boundary of G. The result above is visualized in Figures 1d and 2d.