- Research Article
- Open Access

# On Isoperimetric Inequalities in Minkowski Spaces

- Horst Martini
^{1}Email author and - Zokhrab Mustafaev
^{2}

**2010**:697954

https://doi.org/10.1155/2010/697954

© H. Martini and Z. Mustafaev. 2010

**Received:**11 July 2009**Accepted:**4 March 2010**Published:**28 March 2010

## Abstract

The purpose of this expository paper is to collect some (mainly recent) inequalities, conjectures, and open questions closely related to isoperimetric problems in real, finite-dimensional Banach spaces (= Minkowski spaces). We will also show that, in a way, Steiner symmetrization could be used as a useful tool to prove Petty's conjectured projection inequality.

## Keywords

- Convex Body
- Minkowski Space
- Isoperimetric Inequality
- Intersection Body
- Minkowski Plane

## 1. Introductory General Survey

In Geometric Convexity, but also beyond its limits, isoperimetric inequalities have always played a central role. Applications of such inequalities can be found in Stochastic Geometry, Functional Analysis, Fourier Analysis, Mathematical Physics, Discrete Geometry, Integral Geometry, and various further mathematical disciplines.

We will present a survey on isoperimetric inequalities in real, finite-dimensional Banach spaces, also called *Minkowski spaces*. In the introductory part a very general survey on this topic is given, where we refer to historically important papers and also to results from Euclidean geometry that are potential to be extended to *Minkowski geometry*, that is, to the geometry of Minkowski spaces of dimension
. The second part of the introductory survey then refers already to Minkowski spaces.

### 1.1. Historical Aspects and Results Mainly from Euclidean Geometry

Some of the isoperimetric inequalities have a long history, but many of them were also established in the second half of the 20th century. The most famous isoperimetric inequality is of course the classical one, establishing that among all simple closed curves of given length in the Euclidean plane the circle of the same circumference encloses maximum area; the respective inequality is given by

with being the area enclosed by a curve of length and, thus, with equality if and only if the curve is a circle. In -space the analogous inequality states that if is the surface area of a compact (convex) body of volume , then

holds, with equality if and only if the body is a ball. Note already here that the extremal bodies with respect to isoperimetric problems are usually called *isoperimetrices*.

Osserman [1] gives an excellent survey of many theoretical aspects of the classical isoperimetric inequality, explaining it first in the plane, extending it then to domains in , and describing also various applications (the reader is also referred to [2–4]). In the survey [5] the historical development of the classical isoperimetric problem in the plane is presented, and also different solution techniques are discussed. The author of [6] goes back to the early history of the isoperimetric problem. The paper [7] of Ritoré and Ros is a survey on the classical isoperimetric problem in , and the authors give also a modified version of this problem in terms of "free boundary". A further historical discussion of the isoperimetric problem is presented in [8]. In Chapters 8 and 9 of the book [9] many aspects and applications of isoperimetric problems are discussed, including also related inequalities, the Wulff shape (see the references given there and, in particular, of [10, Chapter ]), and equilibrium capillary surfaces.

Isoperimetric inequalities appear in a large variety of contexts and have been proved in different ways; the occurring methods are often purely technical, but very elegant approaches exist, too. And also new isoperimetric inequalities are permanently obtained, even nowadays. In [11] (see also [12]), the authors prove versions of Petty's projection inequality and the Busemann-Petty centroid inequality (see [13] and below for a discussion of these known inequalities) by using the method of Steiner symmetrization with respect to smooth -projection bodies. In [14] equivalences of some affine isoperimetric inequalities, such as "duals" of versions of Petty's projection inequality and "duals" of versions of the Busemann-Petty inequality, are established; see also [15]. Here we also mention the paper [16], where the method of Steiner symmetrization is discussed and many references are given.

If
is a convex body in
with surface area
and volume
, then for
the Bonnesen inequality states that
, where
is length,
is area, and
and
stand for in- and circumradius of
relative to the Euclidean unit ball (see also the definitions below), with equality if and only if
is a ball. In [17], Diskant extends Bonnesen's inequality (estimating the *isoperimetric deficit*,
, from below) for higher-dimensional spaces. Osserman establishes in [2] the following versions of the isoperimetric deficit in
:

It is well known that a convex -gon with perimeter and area satisfies the isoperimetric inequality . In [18] it is shown that this inequality can be embedded into a larger class of inequalities by applying a class of certain differential equations. Another interesting recent paper on isoperimetric properties of polygons is [19].

In [20] it is proved that if is a simplicial polytope (i.e., a convex polytope all of whose proper faces are simplices) in and is the total -dimensional volume of the -faces of with , then

where and are integers with , with equality if and only if is a regular -simplex.

The authors of [21] study the problem of maximizing for smooth closed curves in , where is again the length of and is an expression of signed areas which is determined by the orthogonal projections of onto the coordinate-planes. They prove that , where is the largest positive number such that is an eigenvalue of the skew symmetric matrix with entries , , and .

An interesting and natural *reverse isoperimetric problem* was solved by Ball (see [22, 23]). Namely, given a convex body
, how small can the surface area of
be made by applying affine, volume-preserving transformations? In the general case the extremal body (with largest surface area) is the simplex, and for centrally symmetric
it is the cube. In [23, Lecture 5] a consequence of the Brunn-Minkowski inequality (see below) involving parallel bodies is discussed, and it is shown how it yields the isoperimetric inequality. Further important results in the direction of reverse isoperimetric inequalities are given in [11, 24]. The latter paper deals with
analogues of centroid and projection inequalities; a direct approach to the reverse inequalities for the unit balls of subspaces of
is given, with complete clarification of the extremal cases.

In [25] the authors prove that if
and
are compact, convex sets in the Euclidean plane, then
with equality if and only if
and
are orthogonal segments or one of the sets is a point (here
denotes the *mixed volume* of
and
, defined below). They also show that
; the equality case is known only when
is a polygon.

### 1.2. The Isoperimetric Problem in Normed Spaces

For (normed or) *Minkowski planes* the isoperimetric problem can be stated in the following way: among all simple closed curves of given *Minkowski length* (= length measured in the norm) find those enclosing largest area. Here the Minkowski length of a closed curve
can also be interpreted as the mixed area of
and the polar reciprocal of the Minkowskian unit circle with respect to the Euclidean unit circle rotated through
. In [26] (as well as in [27]) the *solution of the isoperimetric problem for Minkowski planes* is established. Namely, these extremal curves, called *isoperimetrices*
, are translates of the rotated polar reciprocals as described above. Conversely, the same applies to curves of minimal Minkowski length enclosing a given fixed area.

In [28] it is proved that for the Minkowski metric , where is an integer, the solutions of the isoperimetric problem have the form , and in [29] the particular case of taxicab geometry is studied.

In [30] the following isoperimetric inequality for a convex
-gon
in a Minkowski plane with unit disc
and isoperimetrix
is obtained: if
is the
-gon whose sides are parallel to those of
and which is circumscribed about
, then
, with equality if and only if
is circumscribed about an anticircle of radius
, where
stands for the Minkowskian perimeter and
for area. (An *anticircle* of radius
is any translate of a homothetical copy of
with homothety ratio
.)

In [31] the isoperimetric problem in Minkowski planes is discussed for the case that the isoperimetrix is the polar reciprocal of unit discs related to duals of -spaces.

In [32] some families of smooth curves in Minkowski planes are studied. It is shown that if is a closed convex curve with length enclosing area , and is an anticircle with radius enclosing area , then . This inequality is also extended to closed nonconvex curves.

In [33] star-shaped domains in , presented in polar coordinates by equations of the form , are investigated, with being vector from the unit sphere. The isoperimetric deficit of these domains is estimated for various norms of , where again and denote surface area and volume of the domain and stands for the volume of the standard Euclidean ball.

Since a Minkowski space is a normed space, the given norm defines a usual metric in such a space. In [34] it is proved that if is a rectifiable Jordan curve of Minkowski length , that is, with respect to the Minkowski metric , then there is, up to translation, a centrally symmetric curve such that for all . Also, the isoperimetric problem for rectifiable Jordan curves is solved here. Here encloses the largest area in the class of rectifiable Jordan curves , for any .

In [35] the notion of Minkowski space is extended by considering unit spheres as closed, but in general nonsymmetric hypersurfaces, also called *gauges*. The author gives a suitable definition of volume and applies this definition for solving this generalized form of the isoperimetric problem.

Strongly related to isoperimetric problems, in [36] the lower bound for the geometric dilation of a rectifiable simple closed curve
in Minkowski planes is obtained; note that the *geometric dilation* is the supremum of the quotient between the Minkowski length of the shorter part of
between two different points
and
of it, and the normed distance between these points. In [36] it is proved that for rectifiable simple closed curves in a Minkowski plane
this lower bound is a quarter of the circumference of the unit circle of
, and that (in contrast to the Euclidean subcase) this lower bound can also be attained by curves that are not Minkowskian circles. Furthermore, it is shown that precisely in the subcase of strictly convex normed planes only Minkowskian circles can reach that bound. If
split
into two parts of equal Minkowskian lengths, then the normed distance of these points is called *halving distance of*
*in direction*
. In [37] several inequalities are established which show the relation between halving distances of a simple rectifiable closed curve
in Minkowski planes and other Minkowskian quantities, such as minimum width, inradius, and circumradius of
.

Conversely considered, generalized classes of isoperimetric problems in higher-dimensional Minkowski spaces refer to all convex bodies of given mixed volume having minimum surface area. In
-dimensional Minkowski spaces,
, there are several notions of surface area and volume, for each combination of which there is, up to translation, a unique solution of the corresponding isoperimetric problem. Again, this convex body is called the respective *isoperimetrix* and also denoted by
; see [38, Chapter
], for a broad representation of the isoperimetric problem in
, and types of isoperimetrices for correspondingly different definitions of surface area and volume. In [39] the stability of the solution of the isoperimetric problem in
-dimensional Minkowski spaces
is verified (see also [40]). Namely, some upper estimate for the term
is obtained when
holds. Here
and
stand for surface area and volume of a convex body
in a Minkowski space
, respectively. In [41] sharpenings of the isoperimetric problem in
are established. For instance, one of them is given by

where is the inner parallel body of relative to at distance (see [42, page 134], for more about inner/outer parallel bodies).

In the recent book [43] one can find a discussion on how to involve the following version of the isoperimetric inequality into the theory of partial differential equations: let be a bounded domain in , and let be a suitable -dimensional area measure of the boundary of . Then

with equality only for the ball. The relation to Sobolev's inequality is also discussed. Another side of isoperimetric inequalities is presented in [44]: namely, the isoperimetric problem for product probability measures is investigated there.

Finally we mention once more that the monograph [38] contains a wide and deep discussion of the isoperimetric problem for different definitions of surface area and volume in higher dimensions, showing (also with many nice figures) that the isoperimetrices for the Holmes-Thompson definition and the Busemann definition given below belong to important classes of convex bodies known as projection bodies (= centered zonoids) and intersection bodies, respectively; see Section 2 for definitions of these notions. Corresponding isoperimetric inequalities are discussed there, too.

We will continue by discussing recently established isoperimetric inequalities for Minkowski spaces more detailed, also in view of their applications, and we will also pose related conjectures and open questions. Our attention will be restricted to affine isoperimetric inequalities in Minkowski spaces; we will almost ignore (with minor exceptions) asymptotic affine inequalities.

## 2. Definitions and Preliminaries

Recall that a *convex body*
is a compact, convex set with nonempty interior in
, and that
is said to be *centered* if it is symmetric with respect to the origin
of
.

Let
,
, be a
-dimensional real Banach space, that is, a (normed linear or) *Minkowski space* with *unit ball*
, where
is a convex body centered at the origin. The *unit sphere* of
is the boundary of
and denoted by
. The standard Euclidean unit ball of
will be denoted by
, its volume by
, and as usual we denote by
the standard Euclidean unit sphere in
.

Let
be the *Lebesgue measure* induced by the standard Euclidean structure in
. We will refer to this measure as
-dimensional volume (area in
) and denote it by
. The measure
gives rise to consider a dual measure
on the family of convex subsets of the *dual space*
(i.e., the vector space of linear functionals on
, i.e., all linear mappings from
into
with the usual pointwise operations; see [38, Chapter 0]). However, using the standard basis we will identify
and
, and in that case
and
coincide in
. We write
for the
*-dimensional Lebesgue measure* in
, with
, and therefore we simply write
instead of
; again the identification of
and
via the standard basis implies that
and
coincide in
as well. If
, we denote by
the
-dimensional subspace orthogonal to
, and by
the line through the origin parallel to
. By
we denote the usual one-dimensional inner cross-section measure or *maximal chord length* of
in direction
.

One of the well-known inequalities regarding volumes of convex bodies under (vector or) *Minkowski addition*, defined by
for convex bodies
in
, is the *Brunn-Minkowski inequality* which states that, for
,

holds. Here equality is obtained if and only if and are homothetic to each other. In [45], Gardner gives an excellent survey on this inequality, its applications, and extensions.

A Minkowski space
possesses a *Haar measure*
, and this measure is unique up to multiplication of the Lebesgue measure with a positive constant, that is,

Choosing the "correct" multiple, which can depend on orientation, is not as easy as it seems at first glance, but the two measures and have, of course, to coincide in the standard Euclidean space.

For a convex body
in
, we define the *polar body*
of
by

If
is a convex body in
, then the *support function*
of
is defined by

giving the distance from to the supporting hyperplane of with outward normal . Note that if and only if for any .

If
, then its *radial function*
is defined by

giving the distance from to in direction . Note again that if and only if for any . For and any direction these functions satisfy

In view of the latter inequality, we always have .

We mention the relation

between the support function of a convex body and the inverse of the radial function of (see [38, 42, 46, 47] for properties of and results on support and radial functions).

For convex bodies
,
in
we denote by
their *mixed volume*, defined by

with
being *mixed surface area element* of
; see [38, 42, 46–48] for many interesting properties of mixed volumes.

Note that we have if , that if , and that Furthermore, we will write instead of .

We would also like to mention *Steiner's formula for mixed volumes* (see, e.g., [42, Section
]), given by

*Minkowski's inequality for mixed volumes* states that if
and
are convex bodies in
, then

with equality if and only if and are homothetic (see [38, 42, 46–48]). If is the standard unit ball in , then this inequality becomes the standard isoperimetric inequality.

Another inequality referring to mixed volumes is the *Aleksandrov-Fenchel inequality*, stating that for convex bodies
,
in

holds. Here one has equality if and are homothetic. In general, the equality case is still an open question (see [42, Section ]).

If
is a convex body in
, then the *projection body*
of
is defined via its support function by

for each
, where
is the orthogonal projection of
onto
, and
is called the
-*dimensional outer cross-section measure* or *brightness* of
at
. We note that any projection body is a centered zonoid, and that for centered convex bodies
the equality
implies
; see [42, 47] for more information about projection bodies. (*Zonoids* are the limits, in the Hausdorff sense, of *zonotopes*, i.e., of vector sums of finitely many line segments.)

The *intersection body*
of a convex body
in
is defined via its radial function by

for each . Note that if and are centered convex bodies in , then from it follows that (see [47, 49]).

We should also say that any projection body is dual to some intersection body, and that the converse is not true. The reader can also consult the book [50] of Koldobsky about a Fourier analytic characterization of intersection bodies.

Let
and
be convex bodies in
. Then the *relative inradius*
and the *relative circumradius*
of
with respect to
are defined by

respectively.

## 3. Surface Areas, Volumes, and Isoperimetrices in Minkowski Spaces

As already announced, there are different definitions of measures in higher-dimensional Minkowski spaces (see [38, 51, 52], but also [53] for a variant). We define now the most important ones.

Definition 3.1.

Definition 3.2.

Note that these definitions coincide with the standard notion of volume if the space is Euclidean, and that

Let
be a surface in
with the property that at each point
of
there is a unique tangent hyperplane, and that
is the unit normal vector to this hyperplane at
. Then the *Minkowski surface area* of
is defined by

For the *Holmes-Thompson surface area*, the quantity
is defined by

For the *Busemann surface area*,
is defined by

If
is a convex body in
, then the *Minkowski surface area* of
can also be defined by

where
is that convex body whose support function is
The convex body
plays the central role regarding the solution of the *isoperimetric problem* in Minkowski spaces; see again [38] and the definitions below. Recall once more that in two-dimensional Minkowski spaces
is the polar reciprocal of
with respect to the Euclidean unit circle, rotated through
(see [38, 54–56]).

For the Holmes-Thompson measure, is defined by

and therefore a centered zonoid. For the Busemann measure we have

Among the homothetic images of
we want to specify a unique one, called the *isoperimetrix*
and determined by
(see [38]).

Definition 3.3.

Definition 3.4.

## 4. Inequalities in Minkowski Spaces

One of the fundamental theorems in geometric convexity refers to the *Blaschke-Santaló inequality* and states that if
is a centrally symmetric convex body in
, then

with equality if and only if is an ellipsoid. See also [57, 58] for some new results in this direction.

The sharp lower bound on the product
is known only for certain classes of convex bodies, for example, yielding the *Mahler-Reisner Theorem*. This theorem states that if
is a zonoid in
, then

with equality if and only if is a parallelotope. Mahler proved this inequality for , and Reisner established it for the class of zonoids (see [59]). In [60], Saint-Raymond established this inequality for convex bodies with hyperplanes of symmetry whose normals are linearly independent.

In [61] it is proved that there is a constant , independent of , such that . In recent years, there have been attempts to extend the Mahler-Reisner Theorem to all convex bodies (see, e.g., [62, 63]).

If is a convex body in , then

holds, with equality on the right side if and only if is an ellipsoid, and with equality on the left side if and only if is a simplex.

The right inequality is called *Petty's projection inequality*, and the left one was established by Zhang (see [47, 64]).

The following question has been raised several times and is still open (see [65]).

Problem 1.

What is the sharp lower bound on , when is a centrally symmetric convex body in

In [66], it was conjectured that this sharp lower bound is attained when is a parallelotope.

In [67] (see also [68]), Schmuckenschläger defines the convolution square of as the convolution of the indicator function of and , and the distribution function of this convolution is defined by

Based on this, Schmuckenschläger proves that if is a convex body in , then

Furthermore, he proves the following version of Petty's projection inequality: if is a convex body in such that , then

Another proof of this inequality is given in [69]. In this proof one has to take
random segments in
and to consider then their Minkowski average
(recall that the *Minkowski average* of the segments
with
is the zonotope defined by
. Then it is shown that, for
fixed, the supremum of
is minimal for
an ellipsoid. This result implies Petty's projection inequality referring to
.

Setting in Petty's projection inequality, one obtains

with equality if and only if is an ellipsoid (see also [38]).

with equality if and only if is an ellipsoid; see [70]. In [71] (see also [13]) Lutwak says that this conjectured inequality is one of the major open problems in the field of affine isoperimetric inequalities. In [72], Schneider discusses applications of this conjecture in Stochastic Geometry. In [73] (see also [74]) Brannen proves that this inequality holds for 3-dimensional convex cylindrical bodies.

In [75] it is proved that Petty's conjectured projection inequality is equivalent to another open problem (namely the isoperimetric problem for the Holmes-Thompson measure) over the class of origin-symmetric convex bodies, since the following statement is proved there.

Theorem 4.1.

and equality holds if and only if is an ellipsoid.

Similar to (4.7), we state the following conjecture that would also follow from Petty's conjectured projection inequality (see [75]).

Conjecture 4.2.

with equality if and only if is an ellipsoid.

This conjecture as well as Petty's conjectured projection inequality would easily solve the following problem (see also [75, 76]).

Problem 2.

Also in [75] it is shown that if and only if is an ellipsoid.

Furthermore, the affirmative answer of the following question would solve this ratio problem as well.

Problem 3.

We should also mention that for the sharp bounds on are still unknown, thus yielding a challenging open problem. Thompson (private communication) informed us to have a proof that the sharp lower bound on for equals in the case when is either a rhombic dodecahedron or its dual, that is, a cuboctahedron in .

Since the quantity is not changed under dilation, we obtain, setting in Petty's conjectured projection inequality, the following version of this conjecture which is similar to (4.6).

Conjecture 4.3.

with equality if and only if is an ellipsoid.

For the class of centered convex bodies this conjecture would follow from the following question which involves Steiner symmetrization. Recall that if
is a unit vector, the *Steiner symmetral* St
of a convex body
with respect to the hyperplane
is the convex body obtained as union of all translates of chords of
parallel to
, where these chords are translated in their own affine hull such that, in their final position, they intersect
at their midpoints. The respective procedure is usually called *Steiner symmetrization*.

Problem 4.

with equality for each hyperplane through the origin if and only if is an ellipsoid?

Remark 4.4.

Since is a centered convex body, it suffices to take this hyperplane to be .

Recall that Steiner symmetrization does not change the volume of a given convex body. In addition we note that there is a sequence of convex bodies obtained from a given convex body by finitely many successive Steiner symmetrizations such that this sequence converges to an ellipsoid (see [9]). Also, the following interesting property of Steiner symmetrization should be noticed (see [9, 77, 78]).

Proposition 4.5.

From this proposition it follows that and , unless is an ellipsoid.

Favard's Theorem (see [79] or [80]) states that
holds if and only if
is a
-tangent body of a convex body
. (Recall that a convex body
is a
-*tangent body* of a convex body
if and only if through each boundary point of
there exists a supporting hyperplane of
that also supports
; see [46, page 19] or [42, pages 75-76 and 136], for the definition of tangent bodies.)

Setting and in Favard's Theorem, we obtain . Hence if and only if is a -tangent body of .

In [81], Thompson shows that if the unit ball of a Minkowski space is an affine regular rhombic dodecahedron, then Thus, if is an affine regular rhombic dodecahedron in , then and . Furthermore, this is a counterexample to Problem 7.4.2 posed in [38].

Finding the sharp lower bound on is still an open question.

*Busemann's intersection inequality* states that if
is a convex body in
, then

with equality if and only if is an ellipsoid; see [82].

In [83] it is also proved that Busemann's intersection inequality cannot be strengthened to

when is an affine regular rhombic dodecahedron in .

We should also mention that sharp bounds on and for some cases are known. Namely, it is known that with equality on the left if and only if is a cube or cross polytope, and on the right if and only if is an ellipsoid; see [38, 84]. In , for we have with equality if and only if is a regular hexagon (see [38, 76]). In , holds with equality if and only if is a parallelotope (see [84]).

Also, the relations or , with equality if and only if is an ellipsoid, play an essential role for the proof of a conjecture of Rogers and Shephard (given in [84]), leading to the following theorem.

Theorem 4.6.

Furthermore, equality for each holds if and only if is an ellipsoid.

One could also raise the following question.

Problem 5.

for each .

Our guess is that such a body does not exist.

## Declarations

### Acknowledgments

The authors wish to thank the referees for their valuable comments and suggestions. The second author thanks the Faculty of Mathematics, University of Technology Chemnitz, for hospitality and excellent working conditions. He also thanks the University of Houston-Clear Lake for its support via the FRSF Award no. 970.

## Authors’ Affiliations

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