Metrics and Balls

Question 1

For any metric space (X, d), the open ball and closed ball of radius r > 0 around a point x ∈ X

Answer 1

Br(x) := {y : d(y, x) < r} and

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Br(x) := {y : d(y, x) ≤ r} ⊆ X

Question 2

Euclidean metric open and closed balls

Answer 2

Br(x) = {y ∈ R^n: SUM [ (yi-xi)^2 ] ^1/2 < r}

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Br(x) = (y ∈ R^n: SUM [ (yi-xi)^2 ] ^1/2 ≤ r}

Question 3

Taxicab metric d1 on R^n

Answer 3

d_1(x, y) = |x1 − y1| + · · · + |xn − yn|

Question 4

Discrete metric balls

Answer 4

Br(x) =
( {x} if r ≤ 1
( X if r > 1

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Br(x) =
( {x} if r < 1
( X if r ≥ 1

Question 5

Discrete metric

Answer 5

d(x,y) =
( 0 if x=y
( 1 otherwise

Question 6

Standard metric d_C on the complex numbers C

Answer 6

d_C(z, z’) = |z − z’|

where |z| := r denotes the modulus of the complex number z = re^iθ

Question 7

Edge metric e

Answer 7

The edge metric e on the vertex set V of a path connected graph is defined by:

e(u, w) = min_(π(u,w)) L(π(u, w))

Question 8

Edge metric balls

Answer 8

Br(u) = {w ∈ V : π(u, w) exists with < r edges}.

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Br(u) = {w ∈ V : π(u, w) exists with ≤ r edges}

Question 9

Word Metric

Answer 9

The word metric dw on W is the edge metric on the associated word graph.

ex:
dw(act,boat) = 3
act, bact, baot, boat

Question 10

Metric d_min

Answer 10

d_min(x, y) =

(0 if x = y
(1/(2^n) if n = min{m : x_m != y_m}

Question 11

Metric d∗

Answer 11

d∗(x, y) =

∞ |xj − yj|
SUM [ _____ ]
j=0 2^j

Question 12

d_min balls

Answer 12

B_(1/2^r) (x) = {y ∈ X : y_j = x_j for j ≤ r}

for any integer r ≥ 0
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B_(1/2^r) (x)= {y ∈ X : yj = xj
for j ≤ r − 1}      
= B_(1/2r−1 (x)

for any integer r ≥ 0

Question 13

d_sup metric on X

Answer 13

d_sup(f, g) = sup_( x∈[a,b]) |f(x) − g(x)|

Question 14

L_1[a, b] metric on d_1

Answer 14

For any closed interval [a, b] ⊆ R, the metric d1 is known as the L_1 metric. The space (Y, d_1) is denoted by L_1[a, b]

d_1(f, g) = INT[a:b] [ |f(t) − g(t)| ] dt

Question 15

L2 metric on d_2

Answer 15

d_2(f, g) = [ INT[a:b] (f(t) − g(t))^2 dt ]^1/2

Question 16

Interval metric d_H on X

Answer 16

d_H( [a, b], [r, s] ) = max{ |r − a|, |s − b| }

Question 17

l_1 metric d_1 on X

Answer 17

d1( (ai) , (bi) ) = SUM_( i≥0) [ |a_i − b_i | ]

Question 18

l_1 open and closed balls

Answer 18

B_r( (ai) ) = {(bi) : SUM_( i≥0)  [ |ai − b1| ] < r}
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B_r( (ai) ) = {(bi) : SUM_( i≥0)  [ |ai − b1| ] ≤ r}

Question 19

Euclidean metric

Answer 19

d_2(x, y) = ((x1 − y1)^2 + · · · + (xn − yn)^2)^1/2