Presentation at ICOSC 2022

1. Dimensions of metric spaces of vector groups
S. Bau
School of Mathematics, Statistics and Computer Science, University of Kwazulu-Natal,
Pietermaritzburg, South Africa
The 7th International Conference on Optimization, Simulation
and Control, Ulaanbaatar, Mongolia; Jun 20–22 2022

2. Metric Dimensions of Metric Spaces
Let (X, ρ) be a metric space and ∅ ̸= A ⊆ X. If the set A is well
ordered, then
ρ(A, x) = (ρ(a1, x), ρ(a2, x), . . . , ρ(an, x), . . .)
is a real vector of dimension |A|.

3. Metric Dimensions of Metric Spaces
Let (X, ρ) be a metric space and ∅ ̸= A ⊆ X. If the set A is well
ordered, then
ρ(A, x) = (ρ(a1, x), ρ(a2, x), . . . , ρ(an, x), . . .)
is a real vector of dimension |A|.
If
ρ(A, x) = ρ(A, y) ⇒ x = y
then the subset A is said to resolve (X, ρ).

4. Metric Dimensions of Metric Spaces
Let (X, ρ) be a metric space and ∅ ̸= A ⊆ X. If the set A is well
ordered, then
ρ(A, x) = (ρ(a1, x), ρ(a2, x), . . . , ρ(an, x), . . .)
is a real vector of dimension |A|.
If
ρ(A, x) = ρ(A, y) ⇒ x = y
then the subset A is said to resolve (X, ρ).
Define the metric dimension of (X, ρ) to be
β(X) = min{|A| : A ⊆ X resolves X}.
If A resolves (X, ρ) and |A| is minimum then A is called a metric
basis of (X, ρ) and the vector ρ(A, x) may be thought of as metric
coordinates for the element x ∈ X.

5. A basic question is certainly

6. A basic question is certainly
Problem 1

7. A basic question is certainly
Problem 1
For a metric space, determine its metric dimension.

8. A basic question is certainly
Problem 1
For a metric space, determine its metric dimension.
Literature
Blumenthal, L. M., 1953
Theory and applications of distance geometry, Clarendon Press,
Oxford, 1953.

9. A basic question is certainly
Problem 1
For a metric space, determine its metric dimension.
Literature
Blumenthal, L. M., 1953
Theory and applications of distance geometry, Clarendon Press,
Oxford, 1953.
Harary, F. and Melter, R.A., 1976
On the metric dimension of a graph, Ars Comb., 2(1976), 191-195.

10. A basic question is certainly
Problem 1
For a metric space, determine its metric dimension.
Literature
Blumenthal, L. M., 1953
Theory and applications of distance geometry, Clarendon Press,
Oxford, 1953.
Harary, F. and Melter, R.A., 1976
On the metric dimension of a graph, Ars Comb., 2(1976), 191-195.
Slater, P. J., 1975
Leaves of trees, Congr. Numer., 14(1975), 549-559.

11. Metric spaces
Bau, S. and Beardon, A.F., 2013
The metric dimension of metric spaces, Comput. Methods Funct.
Theory 13(2013), 295-305.
In this paper, Beardon and I proved the following:

12. Metric spaces
Bau, S. and Beardon, A.F., 2013
The metric dimension of metric spaces, Comput. Methods Funct.
Theory 13(2013), 295-305.
In this paper, Beardon and I proved the following:
If X is any one of the classical geometric spaces of Lebesgue
dimension n, then β(X) = n + 1.

13. Metric spaces
Bau, S. and Beardon, A.F., 2013
The metric dimension of metric spaces, Comput. Methods Funct.
Theory 13(2013), 295-305.
In this paper, Beardon and I proved the following:
If X is any one of the classical geometric spaces of Lebesgue
dimension n, then β(X) = n + 1.
If X is any Riemann surface then β(X) = 3.

14. Metric spaces
Bau, S. and Beardon, A.F., 2013
The metric dimension of metric spaces, Comput. Methods Funct.
Theory 13(2013), 295-305.
In this paper, Beardon and I proved the following:
If X is any one of the classical geometric spaces of Lebesgue
dimension n, then β(X) = n + 1.
If X is any Riemann surface then β(X) = 3.
If A is an affine subspace of Rn, then β(A) = dim(A) + 1.

15. Metric spaces
Bau, S. and Beardon, A.F., 2013
The metric dimension of metric spaces, Comput. Methods Funct.
Theory 13(2013), 295-305.
In this paper, Beardon and I proved the following:
If X is any one of the classical geometric spaces of Lebesgue
dimension n, then β(X) = n + 1.
If X is any Riemann surface then β(X) = 3.
If A is an affine subspace of Rn, then β(A) = dim(A) + 1.
We characterized metric spaces with metric dimension 1.

16. Metric spaces
Bau, S. and Beardon, A.F., 2013
The metric dimension of metric spaces, Comput. Methods Funct.
Theory 13(2013), 295-305.
In this paper, Beardon and I proved the following:
If X is any one of the classical geometric spaces of Lebesgue
dimension n, then β(X) = n + 1.
If X is any Riemann surface then β(X) = 3.
If A is an affine subspace of Rn, then β(A) = dim(A) + 1.
We characterized metric spaces with metric dimension 1.

17. Metric spaces
Bau, S. and Beardon, A.F., 2013
The metric dimension of metric spaces, Comput. Methods Funct.
Theory 13(2013), 295-305.
In this paper, Beardon and I proved the following:
If X is any one of the classical geometric spaces of Lebesgue
dimension n, then β(X) = n + 1.
If X is any Riemann surface then β(X) = 3.
If A is an affine subspace of Rn, then β(A) = dim(A) + 1.
We characterized metric spaces with metric dimension 1.
Problem 2

18. Metric spaces
Bau, S. and Beardon, A.F., 2013
The metric dimension of metric spaces, Comput. Methods Funct.
Theory 13(2013), 295-305.
In this paper, Beardon and I proved the following:
If X is any one of the classical geometric spaces of Lebesgue
dimension n, then β(X) = n + 1.
If X is any Riemann surface then β(X) = 3.
If A is an affine subspace of Rn, then β(A) = dim(A) + 1.
We characterized metric spaces with metric dimension 1.
Problem 2
If X is an n-dimensional metric manifold, then is it true that
β(X) = n + 1?

19. Metric spaces
Bau, S. and Beardon, A.F., 2013
The metric dimension of metric spaces, Comput. Methods Funct.
Theory 13(2013), 295-305.
In this paper, Beardon and I proved the following:
If X is any one of the classical geometric spaces of Lebesgue
dimension n, then β(X) = n + 1.
If X is any Riemann surface then β(X) = 3.
If A is an affine subspace of Rn, then β(A) = dim(A) + 1.
We characterized metric spaces with metric dimension 1.
Problem 2
If X is an n-dimensional metric manifold, then is it true that
β(X) = n + 1?
This question was mostly resolved by two Persian mathematicians.

20. Monotonicity
Bau, S. and Lei, Y-M., 2019
Bisectors in vector groups over integers, Bull. Aust. Math. Soc.,
100(3)(2019), 353–361.
Let X, Y be metric spaces with distance functions ρX and ρY. Let
f : X → Y be a mapping.

21. Monotonicity
Bau, S. and Lei, Y-M., 2019
Bisectors in vector groups over integers, Bull. Aust. Math. Soc.,
100(3)(2019), 353–361.
Let X, Y be metric spaces with distance functions ρX and ρY. Let
f : X → Y be a mapping.

22. Monotonicity
Bau, S. and Lei, Y-M., 2019
Bisectors in vector groups over integers, Bull. Aust. Math. Soc.,
100(3)(2019), 353–361.
Let X, Y be metric spaces with distance functions ρX and ρY. Let
f : X → Y be a mapping. An injective isometry f : X → Y is also
called an isometric embedding of X in Y, and X may be considered
as an isometric subspace of Y.

23. Monotonicity
Bau, S. and Lei, Y-M., 2019
Bisectors in vector groups over integers, Bull. Aust. Math. Soc.,
100(3)(2019), 353–361.
Let X, Y be metric spaces with distance functions ρX and ρY. Let
f : X → Y be a mapping. An injective isometry f : X → Y is also
called an isometric embedding of X in Y, and X may be considered
as an isometric subspace of Y.
The metric in X is a restriction of that in Y. That is, ρX = ρY|X.

24. Monotonicity
Bau, S. and Lei, Y-M., 2019
Bisectors in vector groups over integers, Bull. Aust. Math. Soc.,
100(3)(2019), 353–361.
Let X, Y be metric spaces with distance functions ρX and ρY. Let
f : X → Y be a mapping. An injective isometry f : X → Y is also
called an isometric embedding of X in Y, and X may be considered
as an isometric subspace of Y.
The metric in X is a restriction of that in Y. That is, ρX = ρY|X.
The following example is one of the main results of the paper by
Bau and Lei (2019).

25. Monotonicity
Bau, S. and Lei, Y-M., 2019
Bisectors in vector groups over integers, Bull. Aust. Math. Soc.,
100(3)(2019), 353–361.
Let X, Y be metric spaces with distance functions ρX and ρY. Let
f : X → Y be a mapping. An injective isometry f : X → Y is also
called an isometric embedding of X in Y, and X may be considered
as an isometric subspace of Y.
The metric in X is a restriction of that in Y. That is, ρX = ρY|X.
The following example is one of the main results of the paper by
Bau and Lei (2019).
Theorem
There exist a metric space Y and an isometric subspace X of Y with
β(X) β(Y).

26. Consider the perimeter of a circle with perimeter length 9t.

27. Consider the perimeter of a circle with perimeter length 9t. The
points x1, x2, x3, y are taken as shown.

28. Consider the perimeter of a circle with perimeter length 9t. The
points x1, x2, x3, y are taken as shown.
x2
x1 x3
y
3t
3t
t
2t

29. Consider the perimeter of a circle with perimeter length 9t. The
points x1, x2, x3, y are taken as shown.
x2
x1 x3
y
3t
3t
t
2t
Let X = {x1, x2, x3} and Y = {x1, x2, x3, y}, with intrinsic path
metric on the perimeter.

30. Consider the perimeter of a circle with perimeter length 9t. The
points x1, x2, x3, y are taken as shown.
x2
x1 x3
y
3t
3t
t
2t
Let X = {x1, x2, x3} and Y = {x1, x2, x3, y}, with intrinsic path
metric on the perimeter. We verify that

31. Consider the perimeter of a circle with perimeter length 9t. The
points x1, x2, x3, y are taken as shown.
x2
x1 x3
y
3t
3t
t
2t
Let X = {x1, x2, x3} and Y = {x1, x2, x3, y}, with intrinsic path
metric on the perimeter. We verify that
X is an isometric subspace of Y;

32. Consider the perimeter of a circle with perimeter length 9t. The
points x1, x2, x3, y are taken as shown.
x2
x1 x3
y
3t
3t
t
2t
Let X = {x1, x2, x3} and Y = {x1, x2, x3, y}, with intrinsic path
metric on the perimeter. We verify that
X is an isometric subspace of Y;
β(X) = 2 and β(Y) = 1.

33. Vector groups
The concept of a Cayley graph is well known.

34. Vector groups
The concept of a Cayley graph is well known.
Let G be a group and S a generating set for G with S−1 = S and
1 ̸∈ S. Define the Cayley graph X = X(G, S) to be with
V(X) = G,

35. Vector groups
The concept of a Cayley graph is well known.
Let G be a group and S a generating set for G with S−1 = S and
1 ̸∈ S. Define the Cayley graph X = X(G, S) to be with
V(X) = G, E(X) = {gh : g, h ∈ G, gh−1
∈ S}.

36. Vector groups
The concept of a Cayley graph is well known.
Let G be a group and S a generating set for G with S−1 = S and
1 ̸∈ S. Define the Cayley graph X = X(G, S) to be with
V(X) = G, E(X) = {gh : g, h ∈ G, gh−1
∈ S}.
The condition S−1 = S implies that the resulting graph is
undirected (the symmetry axiom for a metric: ρ(x, y) = ρ(y, x)),
and the condition 1 ̸∈ S implies that the graph has no loops (the
distance from a point to itself is 0).

37. Vector groups
The concept of a Cayley graph is well known.
Let G be a group and S a generating set for G with S−1 = S and
1 ̸∈ S. Define the Cayley graph X = X(G, S) to be with
V(X) = G, E(X) = {gh : g, h ∈ G, gh−1
∈ S}.
The condition S−1 = S implies that the resulting graph is
undirected (the symmetry axiom for a metric: ρ(x, y) = ρ(y, x)),
and the condition 1 ̸∈ S implies that the graph has no loops (the
distance from a point to itself is 0). The condition that S is a
generating set of G takes into account that the space is connected
(for we study distances).

38. Vector groups
The concept of a Cayley graph is well known.
Let G be a group and S a generating set for G with S−1 = S and
1 ̸∈ S. Define the Cayley graph X = X(G, S) to be with
V(X) = G, E(X) = {gh : g, h ∈ G, gh−1
∈ S}.
The condition S−1 = S implies that the resulting graph is
undirected (the symmetry axiom for a metric: ρ(x, y) = ρ(y, x)),
and the condition 1 ̸∈ S implies that the graph has no loops (the
distance from a point to itself is 0). The condition that S is a
generating set of G takes into account that the space is connected
(for we study distances). A graph X is a metric space X with its
intrinsic path metric.

39. Examples

40. Examples
This is the ring of Gaussian integers as well as the Cayley graph of
the group Z × Z determined by the generating set
S = {u ∈ Z × Z : |u| = 1}.

41. For a finite example, consider the cyclic group
U6 = {z ∈ C : z6
= 1}, S = {z1, z3, z5}
with z1z5 = 1 = z2
3.

42. For a finite example, consider the cyclic group
U6 = {z ∈ C : z6
= 1}, S = {z1, z3, z5}
with z1z5 = 1 = z2
3. Then we have the complete bipartite graph
1
z1
z2
z3
z4 z5

43. For a finite example, consider the cyclic group
U6 = {z ∈ C : z6
= 1}, S = {z1, z3, z5}
with z1z5 = 1 = z2
3. Then we have the complete bipartite graph
1
z1
z2
z3
z4 z5
There are many interesting questions concerning Cayley graphs.

44. For a finite example, consider the cyclic group
U6 = {z ∈ C : z6
= 1}, S = {z1, z3, z5}
with z1z5 = 1 = z2
3. Then we have the complete bipartite graph
1
z1
z2
z3
z4 z5
There are many interesting questions concerning Cayley graphs.
Our current interest is in infinite Cayley graphs as metric spaces.

45. Bisectors
Let X be a metric space with distance function ρ. Let U, V ⊆ X.
Define the bisector of U, V to be the set
B(U, V) = {x ∈ X : ρ(U, x) = ρ(V, x)}.

46. Bisectors
Let X be a metric space with distance function ρ. Let U, V ⊆ X.
Define the bisector of U, V to be the set
B(U, V) = {x ∈ X : ρ(U, x) = ρ(V, x)}.
If U = {u} and V = {v} then we have the bisector of u, v:
B(u, v) = {x ∈ X : ρ(u, x) = ρ(v, x)}.

47. Bisectors
Let X be a metric space with distance function ρ. Let U, V ⊆ X.
Define the bisector of U, V to be the set
B(U, V) = {x ∈ X : ρ(U, x) = ρ(V, x)}.
If U = {u} and V = {v} then we have the bisector of u, v:
B(u, v) = {x ∈ X : ρ(u, x) = ρ(v, x)}.
In a Euclidean space, the bisectors are exactly the Euclidean
bisectors. That is,
B(x, y) = {z : |z − x| = |z − y|}.

48. In the paper by Bau and Lei (2019), it was shown that

49. In the paper by Bau and Lei (2019), it was shown that
For n ≥ 2, the metric space (Zn, S) with S = {u ∈ Zn : |u| = 1}
cannot be finitely resolved. (part of this result was known
before the paper.)

50. In the paper by Bau and Lei (2019), it was shown that
For n ≥ 2, the metric space (Zn, S) with S = {u ∈ Zn : |u| = 1}
cannot be finitely resolved. (part of this result was known
before the paper.)
Bisectors in these spaces are completely determined.

51. In the paper by Bau and Lei (2019), it was shown that
For n ≥ 2, the metric space (Zn, S) with S = {u ∈ Zn : |u| = 1}
cannot be finitely resolved. (part of this result was known
before the paper.)
Bisectors in these spaces are completely determined.

52. In the paper by Bau and Lei (2019), it was shown that
For n ≥ 2, the metric space (Zn, S) with S = {u ∈ Zn : |u| = 1}
cannot be finitely resolved. (part of this result was known
before the paper.)
Bisectors in these spaces are completely determined.
In Bau and Beardon (2013), a basic observation was proved:

53. In the paper by Bau and Lei (2019), it was shown that
For n ≥ 2, the metric space (Zn, S) with S = {u ∈ Zn : |u| = 1}
cannot be finitely resolved. (part of this result was known
before the paper.)
Bisectors in these spaces are completely determined.
In Bau and Beardon (2013), a basic observation was proved:
Let X be a metric space. A subset A ⊆ X does not resolve X if and
only if A is contained in a bisector.

54. In the paper by Bau and Lei (2019), it was shown that
For n ≥ 2, the metric space (Zn, S) with S = {u ∈ Zn : |u| = 1}
cannot be finitely resolved. (part of this result was known
before the paper.)
Bisectors in these spaces are completely determined.
In Bau and Beardon (2013), a basic observation was proved:
Let X be a metric space. A subset A ⊆ X does not resolve X if and
only if A is contained in a bisector.
This observation points our attention to the study of bisectors in
metric spaces.

55. A few known bisectors. ,

56. A few known bisectors. ,
u
v

57. A few known bisectors. ,
u
v
x
y
F
p
2, 0
P(x, y)

58. Bisectors in vector groups
In Bau and Lei (2019), bisectors in vector groups over Z are
determined.

59. Bisectors in vector groups
In Bau and Lei (2019), bisectors in vector groups over Z are
determined. A few examples are shown below.

60. Bisectors in vector groups
In Bau and Lei (2019), bisectors in vector groups over Z are
determined. A few examples are shown below. You can easily verify
that the set of blue points is the bisector between the pair of red
points.

61. Bisectors in vector groups
In Bau and Lei (2019), bisectors in vector groups over Z are
determined. A few examples are shown below. You can easily verify
that the set of blue points is the bisector between the pair of red
points.

62. Bisectors in vector groups
In Bau and Lei (2019), bisectors in vector groups over Z are
determined. A few examples are shown below. You can easily verify
that the set of blue points is the bisector between the pair of red
points.

63. Examples
a
A

64. A proper generalization
Definition

65. A proper generalization
Definition
For each i = 1, 2, . . . , n, . . ., let Ai ⊆ X and
A = {A1, A2, . . . , An, . . .}.
If
x ̸= y ⇒ ∃i ≥ 1, ρ(Ai, x) ̸= ρ(Ai, y),
then the set A is said to resolve X. If A resolves X and the
cardinality |A | is minimum, then the cardinality δ = |A | is called
the (generalized) metric dimension of X, denoted δ(X).

66. A proper generalization
Definition
For each i = 1, 2, . . . , n, . . ., let Ai ⊆ X and
A = {A1, A2, . . . , An, . . .}.
If
x ̸= y ⇒ ∃i ≥ 1, ρ(Ai, x) ̸= ρ(Ai, y),
then the set A is said to resolve X. If A resolves X and the
cardinality |A | is minimum, then the cardinality δ = |A | is called
the (generalized) metric dimension of X, denoted δ(X).
If each subset Ai = {ai} is a singleton, then we have
δ(X) = β(X). Hence δ(X) is a generalization of metric
dimension.

67. A proper generalization
Definition
For each i = 1, 2, . . . , n, . . ., let Ai ⊆ X and
A = {A1, A2, . . . , An, . . .}.
If
x ̸= y ⇒ ∃i ≥ 1, ρ(Ai, x) ̸= ρ(Ai, y),
then the set A is said to resolve X. If A resolves X and the
cardinality |A | is minimum, then the cardinality δ = |A | is called
the (generalized) metric dimension of X, denoted δ(X).
If each subset Ai = {ai} is a singleton, then we have
δ(X) = β(X). Hence δ(X) is a generalization of metric
dimension.
If A is a partition of X, then δ(X) = βp(X), and hence δ(X)
generalizes the partition dimension which was studied for
graphs.

68. Thank you!