On Isoperimetric Inequalities in Minkowski Spaces

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.

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

(1.1)

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

(1.2)

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 :

(1.3)

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

(1.4)

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

(1.5)

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

(1.6)

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 ,

(2.1)

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,

(2.2)

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

(2.3)

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

(2.4)

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

(2.5)

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

(2.6)

In view of the latter inequality, we always have .

We mention the relation

(2.7)

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

(2.8)

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

(2.9)

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

(2.10)

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

(2.11)

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

(2.12)

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

(2.13)

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

(2.14)

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.

If is a convex body in , then the -dimensional Holmes-Thompson volume of is defined by

(3.1)

Definition 3.2.

If is a convex body in , then the -dimensional Busemann volume of is defined by

(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

(3.3)

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

(3.4)

For the Busemann surface area, is defined by

(3.5)

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

(3.6)

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

(3.7)

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

(3.8)

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

Definition 3.3.

The isoperimetrix for the Holmes-Thompson measure is defined by

(3.9)

Definition 3.4.

Theisoperimetrix for the Busemann measure is defined by

(3.10)

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

(4.1)

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

(4.2)

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

(4.3)

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

(4.4)

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

(4.5)

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

(4.6)

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

(4.7)

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

Petty's conjectured projection inequality states that if is a convex body in with , then

(4.8)

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.

Let be the unit ball of . Then Petty's conjectured projection inequality is true for if and only if

(4.9)

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.

If is the unit ball of , then we have

(4.10)

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.

Let be the unit ball of . Is it then true that

(4.11)

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.

Let be a centered convex body in . Is it then true that

(4.12)

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.

If is a convex body in with , then

(4.13)

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.

Let be a centered convex body in . Is it true that the Steiner symmetral of , created with respect to a given hyperplane through the origin, satisfies the inequality

(4.14)

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.

Let be a centered convex body in . Then the Steiner symmetral of with respect to a given hyperplane through the origin satisfies the inequality

(4.15)

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

(4.16)

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

(4.17)

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.

If is a convex body in , then there exists a direction such that

(4.18)

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

One could also raise the following question.

Problem 5.

Does there exist a centered convex body in such that

(4.19)

for each .

Our guess is that such a body does not exist.