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α 2
α 1 2
= α p
where β =
and P(·) and F(·) are defined in Theorem 2.28. In
3-D it is defined as:
area 3
volume 2
(4r 2 G(β,γ)) 3
( 3 r 3 T(β,γ) 2
6(G(β,γ)) 3
(T(β,γ)) 2
ψ 3 (β,γ)
=
=
=
α 1 2 3 = α 2 3
α 2 3
α 1 2 3 = α p , and G(·,·) and T(·,·) are
α 3
where β =
, γ =
p
defined in Theorem 2.29.
2.5.6 Hyperspheres of Weighted t-Cost Distance
The generalized computations of the vertices and volume of a hypersphere
H(w;r) of weighted t-cost norms WD n (w) has so far been elusive for arbitrary
cost vectors w. However, there are interesting results for well-behaved cost
vectors. A cost vector w is said to be well-behaved if the weights are ordered in
non-increasing manner. That is, w 1
≥ 0. The corresponding
distance is known as awell-behaved weighted t-cost distance.
For a well-behaved w, we first compute the vertices of a hypersphere
H(w;r) following Lemma 3 in [150]. The lemma has some limitations, as
it does not apply the constraint on non-increasing order of the coordinates
of the vertices, which is assumed in the proposition. Hence we present its
modified version in the following theorem. The proof follows the arguments
presented in [150].
≥w 2
≥···≥w n
Theorem 2.30. For a well-behaved w, vertices of H(w;r) are given by φ(u)
where u ∈ Σ n and
8
<
: w i
for i = 1.
0
1
i−1
u i =
@
A
r
w i
min
u j ,u i−1
2 ≤ i≤ n
j=1
The result reported previously (Lemma 3 of [150]) is a special case of the
above lemma. This is stated in the following corollary.
1
w 1
1
w 2
w 1
Corollary 2.12 (Lemma 3 of [150]). If
≥ ... ≥
1
w n− 1
w n
≥ 0, the vertices of H(w;r) are given by φ(u) where u ∈ Σ n
and
r
w i
for i = 1.
u i =
1
w i
1
w i− 1
r 2 ≤ i≤ n
t .
Hence, the vertices of the hypersphere H isr (r) of the inverse square root
weighted t-cost norm (WD isr (·)) are obtained as:
The above corollary holds for inverse square root weights, that is, w t =
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