Optimality of Estimates for the Width of Support Layers of the Isoperimetrix in the Minkowski Geometry
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irk-123456789-1066522016-10-02T03:02:51Z Optimality of Estimates for the Width of Support Layers of the Isoperimetrix in the Minkowski Geometry Diskant, V.I. 2010 Article Optimality of Estimates for the Width of Support Layers of the Isoperimetrix in the Minkowski Geometry / V.I. Diskant // Журнал математической физики, анализа, геометрии. — 2010. — Т. 6, № 4. — С. 396-405. — Бібліогр.: 8 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106652 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Article |
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Diskant, V.I. |
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Diskant, V.I. Optimality of Estimates for the Width of Support Layers of the Isoperimetrix in the Minkowski Geometry Журнал математической физики, анализа, геометрии |
| author_facet |
Diskant, V.I. |
| author_sort |
Diskant, V.I. |
| title |
Optimality of Estimates for the Width of Support Layers of the Isoperimetrix in the Minkowski Geometry |
| title_short |
Optimality of Estimates for the Width of Support Layers of the Isoperimetrix in the Minkowski Geometry |
| title_full |
Optimality of Estimates for the Width of Support Layers of the Isoperimetrix in the Minkowski Geometry |
| title_fullStr |
Optimality of Estimates for the Width of Support Layers of the Isoperimetrix in the Minkowski Geometry |
| title_full_unstemmed |
Optimality of Estimates for the Width of Support Layers of the Isoperimetrix in the Minkowski Geometry |
| title_sort |
optimality of estimates for the width of support layers of the isoperimetrix in the minkowski geometry |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2010 |
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http://dspace.nbuv.gov.ua/handle/123456789/106652 |
| citation_txt |
Optimality of Estimates for the Width of Support Layers of the Isoperimetrix in the Minkowski Geometry / V.I. Diskant // Журнал математической физики, анализа, геометрии. — 2010. — Т. 6, № 4. — С. 396-405. — Бібліогр.: 8 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT diskantvi optimalityofestimatesforthewidthofsupportlayersoftheisoperimetrixintheminkowskigeometry |
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2025-07-07T18:49:24Z |
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2025-07-07T18:49:24Z |
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| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2010, vol. 6, No. 4, pp. 396–405
Optimality of Estimates for the Width of Support Layers
of the Isoperimetrix in the Minkowski Geometry
V.I. Diskant
Cherkasy State Technological University
460 Shevchenko Bvld., Cherkasy, 18006, Ukraine
E-mail:diskant vi@mail.ru
Received December 29, 2009
For the unit ball B of the Minkowski space Mn in the sense of H. Buse-
mann there are proved necessary and sufficient conditions when the inequa-
lities
∆B(QI(ū)) ≥ 4υn−1
nvn
, ∆B(I) ≥ 4υn−1
nvn
, q(I, B) ≥ 2υn−1
nvn
,
∆B(QI(ū)) ≤ 4υn−1
vn
, DB(I) ≤ 4υn−1
vn
, Q(I, B) ≤ 2υn−1
vn
turn into equalities.
All the estimates are optimal since for every estimate there exists the
corresponding B. Here ∆B(I) and DB(I) denote the width and the diameter
of the isoperimetrix I of Mn, q(I,B) stands for the capacity coefficient of B
with respect to I, Q(I, B) is the inclusion coefficient of I with respect to B,
and ∆B(QI(ū)) denotes the width of the support layer of I orthogonal to
a vector ū. The estimates q(I, B) ≥ 2υn−1
nvn
, Q(I,B) ≤ 2υn−1
vn
were obtained
in [5, p. 196]. As shown in Remark 3, the statements on their optimality
given in [5, p. 196] differ from our results.
Key words: isoperimetrix, Minkowski geometry, support layer.
Mathematics Subject Classification 2000: 52A38, 52A40.
The article is devoted to the Minkowski geometry. Let us recall some basic
concepts.
A convex body in the n-dimensional Euclidean space Rn, n ≥ 2, is a convex
compact set with a nonempty interior. Let B be a convex body in Rn, and o be
a point inside B. For an arbitrary point x 6= o, consider a ray in Rn starting
from o and passing through x. Denote by x0 a point at which this ray intersects
the boundary of B.
H. Minkowski [1, p. 26] introduces a distance function in Rn by the formula
g(x̄) = |x̄|
|x̄0| , g(ō) = 0, where x̄ is a position vector of x with respect to o.
c© V.I. Diskant, 2010
Optimality of Estimates for the Width of Support Layers of the Isoperimetrix
Using the pair (B, o), H. Minkowski defines a new distance ρB(x, y) = g(ȳ−x̄)
evaluated for any ordered pair of points x, y in Rn [1, p. 28] and shows that this
distance in Rn generically possesses all the standard properties of the metric
except the property of symmetry.
The geometry of the space obtained by equipping Rn with the new distance
ρB(x, y) is called the Minkowski geometry. The body B is called the unit ball, the
boundary of B is called the etalon surface of measure in the Minkowski geometry
[1, p. 28].
Given a compact A, its surface area S(A,B) is defined in the relative dif-
ferential geometry, corresponding to the Minkowski geometry, by the following
formula [1, p. 80]:
S(A,B) = lim
λ→0
V (A + λB)− V (A)
λ
= nV1(A,B), (1)
here A + λB, λ ≥ 0 stands for a linear combination of A and B in the sense
of Minkowski, V (A) is the volume of A in Rn, V1(A, B) denotes the first mixed
volume of A and B in Rn.
Clearly, the equality ρB(x, y) = ρB(y, x) holds for any pair of points x, y ∈ Rn
if and only if B is centrally symmetric and o is the center of symmetry of B. In this
case ρB(x, y) possesses all the standard properties of the metric. Such a measure
is used by H. Minkowski in the number theory [1, p. 28].
Now let B be a centrally symmetric convex body in the n-dimensional affine
space (n ≥ 2), and o be the center of symmetry of B. Consider a Minkowski
metric in An setting ρB(x, y) = g(ȳ − x̄), where g(x̄) = x̄
x̄0
. Thus, we obtain an
n-dimensional Minkowski space Mn, n ≥ 2 [2, p. 114] with the unit ball B.
Introduce an auxiliary Euclidean structure in Mn. Fix a coordinate system in
Mn, whose origin point coincides with o, and fix an inner product in Mn with the
help of some positive determined symmetric bilinear form. This inner product
generates an auxiliary Euclidean metric in Mn.
Following the H. Busemann ideas [2, p. 278], for an arbitrary convex com-
pact A, which belongs to an m-dimensional plane Mm, 1 ≤ m ≤ n, define its
m-dimensional volume V B
m (A) in Mn by the following formula:
V B
m (A) =
Vm(A)
Vm(B ∩Mm
0 )
υm, (2)
where Vm stands for the m-dimensional Lebesgue measure with respect to the
auxiliary Euclidean metric, Mm
0 is the plane passing through o which is parallel
to Mm, and υm denotes the volume of the standard Euclidean unit ball in Rm.
It follows from (2) that V B
n (B) = υn. If the auxiliary Euclidean metric in Mn
is normalized in such a way that
Vn(B) = υn, (3)
Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 4 397
V.I. Diskant
then the volume V B
n (A) is equal to the Euclidean volume Vn(A) for any convex
compact A in Mn.
To define the surface area SB(A) of a convex compact A in Mn, H. Busemann
considers a convex body I in Mn, namely the isoperimetrix of Mn. This body is
defined via its support function in the auxiliary Euclidean metric satisfying (3).
The support function hI(ū) of I is represented by the formula
hI(ū) =
υn−1
Vn−1(B ∩ T0(ū))
, ū ∈ Ω, (4)
where Ω is the unit (with respect to the auxiliary Euclidean metric) sphere in
Rn with the center o, T0(ū) is the hyperplane passing through o orthogonal to ū
[2, p. 180].
In [2, p. 182], H. Busemann shows that if m = n− 1, then by (2) the surface
area SB(A) of any compact A in Mn with respect to the auxiliary Euclidean
metric verifying (3) satisfies the following equality:
SB(A) = nV1(A, I). (5)
It is also shown in [2, p. 279] that I depends only on the unit ball B of Mn
and does not depend on the auxiliary Euclidean metric satisfying (3).
Given a convex body A in Mn, let TA be an arbitrary support hyperplane of
A, T ′A be a support hyperplane of A parallel to TA and not coinciding with TA.
The point set Q(TA) = TA ∩ T ′A, where TA denotes the closed support half-
space of A bounded by TA, is called the support layer corresponding to TA.
The width of Q(TA) is defined by the formula ∆B(Q(TA)) = 2q(Q(TA), B), where
q(Q(TA), B) stands for the capacity coefficient of B with respect to Q(TA), i.e.
such a maximum of α that αB can be placed into Q(TA) by some translation.
If Mn is provided with some auxiliary Euclidean metric, then it is natural to
denote the support layer Q(TA) by QA(ū), where ū ∈ Ω is a unit vector orthogonal
to the support hyperplanes TA and T ′A of the body A.
The following statements on the support layers width were obtained in [3,
pp. 390, 391].
Theorem 1. If ū ∈ Ω is an arbitrary unit vector in Mn with an arbi-
trary auxiliary Euclidean metric, then the width ∆B(QA(ū)) of the corresponding
support layer QA(ū) holds the equation
∆B(QA(ū)) = 2q(QA(ū), B) = 2
hA(ū) + hA(−ū)
hB(ū) + hB(−ū)
, (6)
where hA(ū) is a support value of A corresponding to ū in the used Euclidean
metric.
398 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 4
Optimality of Estimates for the Width of Support Layers of the Isoperimetrix
R e m a r k 1. It is shown in [3, p. 390] that the right-hand side of (6) is
continuous and it has a maximum and a minimum in Ω. The maximal value of (6)
denoted by DB(A) is called the diameter of the body A in Mn. The minimal value
of (6) denoted by ∆B(A) is called the width of the body A in Mn. The diameter
DB(A) is proved [4, p. 220] to be the maximal distance between points of A
in Mn.
R e m a r k 2. Since B and I are centrally symmetric convex bodies in Mn,
with point o being a joint center of symmetry, then it follows from Theorem 1
that for any choice of the auxiliary Euclidean metric in Mn the width of support
layers of the isoperimetrix I is represented by the following formula:
∆B(QI(ū)) = 2
hI(ū)
hB(ū)
.
Theorem 2. The support layer width ∆B(QI(ū)) of the isoperimetrix I in
Mn satisfies the following estimates:
4υn−1
nυn
≤ 2
hI(ū)
hB(ū)
≤ 4υn−1
υn
. (7)
Theorem 3. The width ∆B(I) and the diameter DB(I) of the isoperimetrix
I in Mn satisfy the following estimates:
4υn−1
nυn
≤ ∆B(I) ≤ DB(I) ≤ 4υn−1
υn
. (8)
Let q(I, B) denote the capacity coefficient of the unit ball B with respect to
the isoperimetrix I in Mn, i.e. such a maximal α that the body αB can be placed
into I by some translation. Let Q(I,B) denote the inclusion coefficient of the
isoperimetrix I with respect to the unit ball B, i.e. such a minimal β that I can
be placed into βB by some translation.
The aim of this article is to prove the following results.
Theorem 4. The following equalities hold:
∆B(I) = 2q(I, B), DB(I) = 2Q(I, B).
Theorem 5. Let Mn be provided with an auxiliary Euclidean metric satisfy-
ing (3). Then the inequality
2υn−1
nυn
≤ hI(ū)
hB(ū)
(9)
Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 4 399
V.I. Diskant
turns into equality for the unit ball B and a vector ū0 ∈ Ω in the space Mn if and
only if the Schwarz symmetrization [2, p. 224] of the unit ball B with respect to
a straight line with direction ū0 ∈ Ω results into a right circular bicone.
Theorem 6. Let Mn be provided with an auxiliary Euclidean metric satisfy-
ing (3). Then the inequalities
4υn−1
nυn
≤ ∆B(I), (10)
2υn−1
nυn
≤ q(I, B) (11)
turn into equalities for the unit ball B of Mn if and only if there exists a unit
vector ū0 ∈ Ω in Mn such that the Schwarz symmetrization of the unit ball B
with respect to a straight line with direction ū0 results into a right circular bicone.
Theorem 7. Let Mn be provided with an auxiliary Euclidean metric satisfy-
ing (3). Then the inequality
hI(ū)
hB(ū)
≤ 2υn−1
υn
(12)
turns into equality for the unit ball B and a vector ū0 ∈ Ω in the space Mn if and
only if the Schwarz symmetrization of the unit ball B with respect to a straight
line with direction ū0 ∈ Ω results into a right circular cylinder.
Theorem 8. Let Mn be provided with an auxiliary Euclidean metric satisfy-
ing (3). Then the inequalities
DB(I) ≤ 4υn−1
υn
, (13)
Q(I, B) ≤ 2υn−1
υn
(14)
turn into equalities for the unit ball B of the space Mn if and only if there exists
a unit vector ū0 ∈ Ω in Mn such that the Schwarz symmetrization of the unit ball
B with respect to a straight line with direction ū0 ∈ Ω results into a right circular
cylinder.
Theorem 9. If the unit ball B of Mn, n ≥ 3, is a parallelepiped and V (B) =
υn, then
∆B(I) >
2υn−1
nυn
.
Theorem 10. If the unit ball B of Mn is centrally symmetric, Vn(B) = υn,
and A is a convex body in Mn, then
2υn−1
nυn
≤ SB(A)
S(A,B)
≤ 2υn−1
υn
, (15)
2υn−1 ≤ SB(B) ≤ 2nυn. (16)
400 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 4
Optimality of Estimates for the Width of Support Layers of the Isoperimetrix
R e m a r k 3. Let B be a unit ball of Mn in the sense of H. Busemann.
In [5, p. 190], a convex body whose support function with respect to an auxiliary
Euclidean metric has the form (4) is denoted by IB and the isoperimetrix of Mn
which satisfies ÎB = V (B)
υn
IB is denoted by ÎB. If the assumption (3) is satisfied,
then ÎB just coincides with the isoperimetrix I of Mn as viewed in our article.
In [5, p. 190], for an arbitrary convex body K in Mn, an intrinsic radius r(K)
and an extrinsic radius R(K) of K in Mn are defined as follows:
r(K) = max
{
α : ∃x ∈ Mn such that αÎB ⊆ K + x
}
,
R(K) = min
{
α : ∃x ∈ Mn such that αÎB ⊇ K + x
}
.
Following our notations, one has r(K) = q(K, I), R(K) = Q(K, I).
Since the equality q(A,C) = 1
Q(C,A) holds for both convex bodies A and C in
Rn [6, p.102], then
r(K) =
1
Q(I,K)
,
R(K) =
1
q(I,K)
.
The following estimates for the intrinsic and extrinsic radiuses of B in Mn
were obtained in [5, p. 196]:
r(B) ≥ υn
2υn−1
, (17)
R(B) ≤ nυn
2υn−1
. (18)
Comparing our results with those obtained in [5], one can see that the estimate
(17) is equivalent to (14) since r(B) = 1
Q(I,B) . Moreover, the estimate (18) is
equivalent to (11) since R(B) = 1
q(I,B) .
It is stated in [5, p. 196] that the estimate (17) is not optimal, whereas the
estimate (18) turns into equality if and only if B is a parallelotop.
In fact, the estimate (17) is optimal since it is equivalent to (14). The condi-
tions necessary and sufficient for the estimate (14) (as well as for (17)) to turn
into equality are given in Theorem 8. For example, the equality in (17) holds if
B is a cube in Mn and ū0 is orthogonal to a cube face.
The estimate (18) is actually optimal. However, if B is a parallelepiped, as
well as a parallelotop, then it follows from Theorem 9 that the estimate (18) does
not turn into equality in the case of n ≥ 3. The conditions necessary and sufficient
Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 4 401
V.I. Diskant
for the estimate (18) to turn into equality are given in Theorem 6. For example,
the estimate (18) turns into equality if B is a right circular bicone in Mn and ū0
is orthogonal to the base of bicone.
P r o o f o f T h e o r e m 4. Let us show that
∆B(I) ≤ 2q(I,B). (19)
From the definition of ∆B(I) we have ∆B(I) = min
ū∈Ω
2 hI(ū)
hB(ū) . Therefore,
∆B(I) ≤ 2 hI(ū)
hB(ū) for any ū ∈ Ω. Hence, 1
2∆B(I)hB(ū) ≤ hI(ū) for any ū ∈ Ω.
Taking into account the properties of the support functions (see [1, p. 30]), we
obtain 1
2∆B(I)B ⊂ I. Thus, 1
2∆B(I) ≤ q(I,B).
Now let us take q = q(I, B) and place qB into I by some translation, so
qB1 ⊂ I. Then the inequality
∆B(QqB1(ū)) ≤ ∆B(QI(ū)) (20)
holds for any ū ∈ Ω. Since the width of the support layer orthogonal to ū ∈ Ω is
constant under translation, then it follows from (20) that the inequality
2q ≤ 2
hI(ū)
hB(ū)
(21)
holds, here 2q is the width of the support layer of qB orthogonal to ū, as well
2 hI(ū)
hB(ū) stands for the width of the support layer of I orthogonal to ū in Mn.
Finally, (21) leads to the following inequality:
2q(I,B) ≤ min
ū∈Ω
2
hI(ū)
hB(ū)
= ∆B(I).
This inequality and (19) imply the desired equality stated in Theorem 4, q.e.d.
P r o o f o f T h e o r e m 5. Let Mn be provided with an auxiliary
Euclidean metric satisfying (3). Chose an orthonormal vector basis and denote
by ox1 . . . xn the corresponding Cartesian coordinate system whose origin o is
the center of symmetry of B. The hyperplane xn = 0 divides B into two parts
B1 = B ∩ (xn ≥ 0) and B2 = B ∩ (xn ≤ 0). Clearly, V (B1) = V (B2) = υn
2 . Let
[−b, b], b > 0, be the image of B under the orthogonal projection into oxn. Then
hB(ū) = hB1(ū) = b for the unit vector ū ∈ Ω along the axis oxn.
Apply to B the Schwarz symmetrization [2, p. 224] with respect to the axis
oxn and denote the result by B̃. The orthogonal projection of B̃ onto the axis
oxn is the segment [−b, b]. The intersection B̃ ∩ (xn = c), −b ≤ c ≤ b, represents
an (n − 1)-dimensional ball centered at the point xn = c on the axis oxn, its
(n−1)-dimensional volume is equal to Vn−1(B̃∩ (xn = c)) = Vn−1(B∩ (xn = c)).
402 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 4
Optimality of Estimates for the Width of Support Layers of the Isoperimetrix
The bodies B̃1 = B̃ ∩ (xn ≥ 0) and B̃2 = B̃ ∩ (xn ≤ 0) are obtained by the
symmetrization of the bodies B1 and B2, respectively. Moreover, one has V (B̃) =
V (B) = υn, V (B̃1) = V (B1) = υn
2 , hB(ū) = hB1(ū) = hB̃1
(ū) = hB̃(ū) = b.
It is shown in [3, p. 392] that Vn−1(B ∩ (xn = α)) = Vn−1(B ∩ (xn = −α)) ≤
Vn−1(B ∩ (xn = 0)), 0 ≤ α ≤ b. Hence, B̃ is symmetric with respect to the
hyperplane xn = 0.
Given B1 and B̃1, consider the right cone K whose base is B̃1∩ (xn = 0)) and
whose height is the segment [0, b] of the axis oxn.
Let us show that the following equality holds:
hI(ū)
hB(ū)
V (K) =
υn−1
n
. (22)
Express V (K) via hI(ū)
hB(ū) by using (4). One has
V (K) =
1
n
bVn−1(B̃1 ∩ (xn = 0)) =
hB(ū)υn−1
nυn−1
Vn−1(B ∩ (xn = 0))
=
hB(ū)υn−1
nhI(ū)
,
which leads to (22).
Now, if (9) turns into equality for some ū = ū0, i.e., hI(ū0)
hB(ū0) = 2υn−1
nυn
, then (22)
leads to V (K) = υn
2 . Hence, V (K) = V (B̃1). Since K ⊂ B̃1 and the interior of
K is nonempty, then K = B̃1, so B̃ is a right circular bicone.
On the other hand, if the Schwarz symmetrization of B with respect to the
straight line parallel to the vector ū = ū0 results in a bicone, then V (K) =
V (B̃1) = υn
2 . Hence it follows from (22) that (9) turns into equality for ū = ū0,
q.e.d.
P r o o f o f T h e o r e m 6. By definition, the width ∆B(I) of the
isoperimetrix I is equal to ∆B(I) = min
ū∈Ω
2 hI(ū)
hB(ū) .
It follows from Remark 1 that hI(ū)
hB(ū) is defined and continuous in Ω. Hence
it attains its minimum at some ū0 ∈ Ω. Then the equality 4υn−1
nυn
= ∆B(I) is
equivalent to the equality 2υn−1
nυn
= hI(ū0)
hB(ū0) . Now the statement of Theorem 6 about
the optimality of (10) follows directly from Theorem 5. As for the optimality of
(11), Theorem 6 follows from Theorem 4.
P r o o f o f T h e o r e m 7. Given B1 and B̃1, consider the right
circular cylinder Π, whose base is B̃1 ∩ (xn = 0) and whose height is the segment
[0, b] of the axis oxn. Since Π and K have the same base and height, one has
V (Π) = nV (K). Then (22) leads to the following equality:
hI(ū)
hB(ū)
V (Π) = υn−1. (23)
Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 4 403
V.I. Diskant
If (12) turns into equality for some ū = ū0 ∈ Ω, i.e., hI(ū0)
hB(ū0) = 2υn−1
υn
, then it
follows from (23) that V (Π) = υn
2 . Hence, V (Π) = V (B̃1). Since B̃1 ⊂ Π and the
interior of B̃1 is nonempty, then B̃1 = Π. So, B̃1 is a right circular cylinder and
consequently B̃ is a right circular cylinder.
On the other hand, if B̃1 is a right circular cylinder, then V (Π) = V (B̃1).
Hence V (B̃1) = V (Π) = υn
2 , so (23) leads to hI(ū0)
hB(ū0) = 2υn−1
υn
, q.e.d.
P r o o f o f T h e o r e m 8 is similar to the proof of Theorem 6.
P r o o f o f T h e o r e m 9. Fix a coordinate system so that the origin o is
a vertex of the parallelepiped B and the basis vectors are directed along the edges
of B erected at o. Instead of the initial auxiliary Euclidean metric satisfying (3),
when B is viewed as a a parallelepiped, let us consider a new auxiliary Euclidean
metric satisfying (3), which views B as a cube. It is shown in [2, p. 279] that
the changing of metrics does not change the isoperimetrix, i.e. the isoperimetrix
I of the parallelepiped coincides with the isoperimetrix I of the cube.
The edges of the cube B erected at o are pairwise orthogonal, they have the
same length α, moreover, αn = υn. Consider a new Cartesian coordinate system,
whose origin o1 is the center of symmetry of the cube B and whose axes are
parallel to the edges of B. Evidently, the most distant points of B with respect
to o1 are the vertices of B. Hence for any ū ∈ Ω one has hB(ū) ≤ α
√
n
2 , where
α
√
n denotes the diagonal of cube.
In [7], K. Ball proved the maximal (n − 1)-dimensional volume for the in-
tersections of the unit cube with hyperplanes in Rn (n ≥ 2) to be equal to
√
2.
Therefore, Vn−1(B ∩ To1(ū)) ≤ √
2αn−1 for any ū ∈ Ω. Hence, for the width
∆B(QI(ū)) = 2 hI(ū)
hB(ū) , ū ∈ Ω, of support layers of the isoperimetrix I of Mn,
n ≥ 3, one has the following estimate:
2
hI(ū)
hB(ū)
= 2
υn−1
hB(ū) Vn−1(B ∩ To1(ū))
≥ 2υn−1√
n
2 α
√
2αn−1
=
4υn−1√
2nαn
>
4υn−1
nυn
,
since
√
2n < n for n ≥ 3. By the continuity of hI(ū)
hB(ū) at ū ∈ Ω, one has ∆B(I) =
min
ū∈Ω
2 hI(ū)
hB(ū) > 4υn−1
nυn
for n ≥ 3.
P r o o f o f T h e o r e m 10. Rewrite the inequalities (7) as follows:
2υn−1
nυn
hB(ū) ≤ hI(ū) ≤ 2υn−1
υn
hB(ū).
The above and the properties of support functions [1, p. 30] result
2υn−1
nυn
B ⊂ I ⊂ 2υn−1
υn
B. (24)
404 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 4
Optimality of Estimates for the Width of Support Layers of the Isoperimetrix
Let us use an auxiliary Euclidean metric satisfying (3) in Mn. Due to the
homogeneity and monotonicity of the mixed volume [1, p. 49], the inclusions (24)
lead to the following inequalities:
2υn−1
nυn
V1(A,B) ≤ V1(A, I) ≤ 2υn−1
υn
V1(A,B). (25)
Then take into account (1) and (5) to see that the estimates (15) for SB(A)
S(A,B) hold
for an arbitrary centrally symmetric convex body B and an arbitrary convex
body A in Mn.
Set A = B in (15). Then S(B, B) = nV1(B, B) = nV (B) = nυn, here
SB(B) is the surface area of the unit ball B in Mn in the sense of H. Busemann
(5). Substituting these expressions into (15), one obtains the following known
Busemann–Petty estimates for SB(B) in Mn [8, p. 242]:
2υn−1 ≤ SB(B) ≤ 2nυn−1.
The right-hand side estimate turns into equality if and only if B is a paral-
lelotop [8, p. 242].
References
[1] T. Bonnesen and V. Fenchel, Theory of Convex Bodies. FAZIS, Moscow, 2002.
(Russian)
[2] K. Leichtweis, Convex Sets. Nauka, Moscow, 1985. (Russian)
[3] V.I. Diskant, Estimates for Diameter and Didth of the Isoperimetrix in Minkowski
Geometry. — J. Math. Phys., Anal., Geom. 2 (2006), 388–395.
[4] V.I. Diskant, Refinement of an Isodiametral Inequality in Minkowski Geometry. —
Mat. Fiz., Anal., Geom. 1 (1994), 216–226. (Russian)
[5] H. Martini and Z. Mustafaev, Some Applications of Cross-Section Measures in
Minkowski Spaces. — Period. Math. Hungarica 53 (2006), No. 1–2, 185–197.
[6] V.I. Diskant, Refinements of an Isoperimetric Inequality and Stability Theorems
in the Theory of Convex Bodies. — Tr. Inst. Mat. (Novosibirsk). Sovrem. Probl.
Geom. Analiz 14 (1989), 89–132. (Russian)
[7] K. Ball, Cube Slicing in Rn. — Proc. Amer. Math. Soc. 97 (1986), 465–473.
[8] A.C. Thompson, Minkowski Geometry. Encycl. Math. Appl. 63. Cambridge Univ.
Press, Cambridge, 1996.
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