Mean width of regular polytopes

Mean width of regular polytopes and expected maxima of correlated Gaussian variables

Abstract.

An old conjecture states that among all simplices inscribed in the unit sphere the regular one has the maximal mean width. An equivalent formulation is that for any centered Gaussian vector satisfying one has

where are independent standard Gaussian variables. Using this probabilistic interpretation we derive an asymptotic version of the conjecture. We also show that the mean width of the regular simplex with vertices is remarkably close to the mean width of the regular crosspolytope with the same number of vertices. Interpreted probabilistically, our result states that

where is an absolute constant. We also compute the higher moments of the projection length of the regular cube, simplex and crosspolytope onto a line with random direction, thus proving several formulas conjectured by S. Finch. Finally, we prove distributional limit theorems for the length of random projection as the dimension goes to . In the case of the -dimensional unit cube , we prove that

whereas for the simplex and the crosspolytope the limiting distributions are related to the Gumbel double exponential law.

Key words and phrases:
Mean width, intrinsic volumes, regular simplex, regular crosspolytope, maxima of Gaussian processes, random projections, extreme value theory
2010 Mathematics Subject Classification:
Primary, 52A23; secondary, 52A39, 52A20, 60G15, 60G70, 46B06
The work of the third author is supported by the grant RFBR 13-01-00256 and by the Program of Fundamental Researches of Russian Academy of Sciences “Modern Problems of Fundamental Mathematics”.

1. Conjecture on the mean width

1.1. Introduction

The mean width of a compact convex body is the expected length of a projection of this body onto a line with uniformly chosen, random direction. That is, the mean width equals , where

and is uniformly distributed on the unit sphere .

How should points be arranged on the -dimensional unit sphere so as to maximize the mean width of their convex hull? An old conjecture states (see [14, Section 9.10.2]) that the arrangement must be regular.

The mean width is just a multiple of the first intrinsic volume , namely

(1)

see [20, p. 210]. The first intrinsic volume has the advantage of not depending on the dimension of the surrounding space. Hence the conjecture can be formulated as follows:

(2)

where is a regular simplex with vertices inscribed in the sphere , and denotes the convex hull.

This question is surprisingly hard. Several authors [13, 3, 4, 23] assumed the existence of a proof, but the problem is still open. Besides very natural formulation in Convex Geometry this problem is very important in Information Theory, as it is closely related to the the long-standing simplex code conjecture [8].

1.2. Probabilistic statement

The conjecture can be reformulated in terms of Gaussian processes in the following way. Throughout the paper, denotes a standard Gaussian vector in . Consider a compact set . Using the fact that the norm and the direction of are independent, it is not difficult to derive Sudakov’s formula

(3)

(see [21] for details and for a generalization to the infinite-dimensional case, or Theorem 3.1 in the present paper for a more general result). This probabilistic interpretation of the first intrinsic volume allows to reformulate the conjecture as follows.

Proposition 1.1.

For every integer the following two statements are equivalent:

  • One has

    (4)

    and the equality is attained iff are vertices of a regular simplex.

  • For every centered Gaussian vector satisfying

    (5)

    one has

    (6)

    and the equality is attained iff for all .

Proof.

First of all note that

(7)

because there is an -dimensional affine subspace (and hence, an -dimensional sphere of radius at most ) containing . Therefore, we can restate (i) as follows:

(8)

and the equality is attained iff are vertices of a regular simplex centered at the origin. Let be a standard orthonormal basis in . As a realization of such a simplex we can take the convex hull of the points

To see this, note that the -dimensional regular simplex

can be inscribed in an -dimensional sphere of radius centered at . It follows from (3) applied to that

(9)

To any points we associate a centered Gaussian vector such that via

If we agree to identify two Gaussian vectors if they have the same distribution and two tuples and if for all , then this correspondence becomes one-to-one because . It follows from (3) that

The Gaussian vector corresponding to the points satisfies

Taken together, the above considerations show the equivalence of (i) and (ii). ∎

1.3. Asymptotic version of the conjecture

We now show that (2) holds asymptotically.

Theorem 1.2.

For some absolute constant and all ,

Proof.

The first inequality is trivial because we can take to be the vertices of . Replacing by and using (7) we can restate that second inequality as follows: For all ,

Fix . For define Gaussian random variables and note that has zero mean and unit variance. It is known (see, e.g., [7, p. 138]) that

(10)

We provide a proof for the sake of completeness. For one has

Letting yields (10).

On the other hand, it is well-known in the theory of extreme values, see [15, Theorem 1.5.3 on p. 14] and [19], that

(11)

Using (3) and (10), we obtain

Combining this with (9) and (11) gives

as . This proves the claim. ∎

2. Regular simplex and regular crosspolytope

In this section we compare the mean width of the regular simplex to the mean width of the regular -dimensional crosspolytope defined by

Note that both and (which can be considered as a degenerate simplex) have vertices and can be inscribed in . We will show that conjecture (2) is true in this special case, that is . Moreover, we will prove a lower bound which shows that the mean width of is remarkably close to the mean width of .

2.1. Mean width and extreme values

It follows from Sudakov’s formula (3), see also (9), that

(12)
(13)

where we recall that . It is well-known in the theory of extreme values [15, Theorem 1.5.3 on p. 14] that

(14)
(15)

where is any sequence satisfying , for example1

(16)

Note that (15) (together with (14)) expresses the fact that the minimum and the maximum of become asymptotically independent; see [15, Theorem 1.8.3 on p. 28]. Taking the expectation (which is justified by [19]) and noting that the expectation of the Gumbel distribution on the right-hand side of (14) and (15) is the Euler constant , we obtain the large asymptotics

(17)
(18)

These formulas are known; see [2], [11, p. 5], [10, p. 8].

2.2. Comparing and

We are going to show that distance between and is in fact much smaller than the bound implied by (17) and (18). First we state the corresponding probabilistic result.

Theorem 2.1.

If are independent standard Gaussian variables, then

The left hand-side inequality immediately follows from Slepian’s lemma [15, Corollary 4.2.3 on p. 84] because the random vector is uncorrelated, whereas the off-diagonal correlations of are non-positive. The proof of the second estimate will be given in Section 4. Theorem 2.1 together with (12) and (13) implies the following

Corollary 2.2.

For every ,

We now provide a bound which is asymptotically sharper. Its proof will be given in Section 5.

Theorem 2.3.

Let be independent standard Gaussian variables. Then, as , one has

Combining Theorem 2.3 with (12) and (13) yields the following

Corollary 2.4.

As ,

It is possible to obtain further asymptotic terms in (17) and (18), (see, e.g., [15, Eq. (2.4.8) on p. 39]) but it seems that none of these expansions can correctly capture the very small difference between the expectations in Theorems 2.1 and 2.3.

3. Higher moments and limiting distribution of the random width

3.1. Sudakov’s formula for higher moments

Given a convex compact set we denote by the length of the projection of onto a uniformly chosen direction, that is

(19)

where has uniform distribution on the sphere . In this section we study the higher moments of the random variable .

Recall that denotes a random vector having standard normal distribution on . The isonormal Gaussian process is defined by

It is characterized by

(20)

For a compact set define the range of over to be

The next theorem generalizes Sudakov’s formula (3) to higher moments.

Theorem 3.1.

If the set is convex and compact, then

(21)

If, moreover, the set is symmetric with respect to the origin, then

(22)

Tsirelson [22] generalized Sudakov’s formula (3) to all intrinsic volumes. After the acceptance of this paper we have learned that Paouris and Pivovarov extended Tsirelson’s formula to higher moments (see [17, Prop. 4.1]) thereby proving a more general variant of Theorem 3.1.

Proof.

The standard Gaussian vector in can be written as

where and are such that

  1. is a random vector with uniform distribution on the unit sphere in ;

  2. is a random variable having -distribution with degrees of freedom;

  3. and are independent.

It follows that we have the distributional equality

(23)

Taking -th moments of both parts and noting that and are independent, we obtain that

The moments of the -distribution are known. Inserting the value of the moment, we obtain (21) (which holds without the symmetry assumption on ). If the set is symmetric with respect to the origin, then and we obtain (22). ∎

Remark 3.2.

In particular, taking in Theorem 3.1 and noting that the first intrinsic volume is related to the mean width by (1), we recover from (21) Sudakov’s [21] formula

(24)

Note that the symmetry assumption on is not needed in the derivation of (24) because in the last equality we used only that has the same distribution as .

3.2. Applications to regular polytopes

Theorem 3.1 can be used to prove several conjectures on projections of regular polytopes which are due to Finch [10, 11, 12].

Example 3.3.

Let be the -dimensional cube of unit volume. It is easy to see that . Therefore, by (21),

(25)

In particular, taking and noting that we obtain that the mean width is

or, equivalently, , which is well known. The second moment of the projection length is given by

where we have used that and . This formula has been conjectured by Finch [11, p. 9] who established it for by purely geometric arguments [12]. Using (25) it is possible to compute more moments of , for example

where we have used that , , , .

Example 3.4.

For the regular crosspolytope we have and therefore Theorem 3.1 yields

For , this formula was conjectured by Finch in [10, p. 3]; see also [11].

Example 3.5.

For the regular -dimensional simplex , Theorem 3.1 yields

Note that in this formula, is projected onto a random direction in , even though is -dimensional.

It is more natural to state the corresponding formula for (which is a regular simplex with vertices inscribed in ) projected onto a random direction in . As a realization of we take the points

in the hyperplane (which can be identified with ). By (20), the isonormal process on satisfies

so that for its range on we have

Therefore, for the projection length of onto a uniformly chosen random direction in the hyperplane we obtain

For , this formula was conjectured by Finch [11, p. 4] who verified it for small values of .

3.3. Limit distribution for the random width

What is the asymptotic distribution of the projection length of a high-dimensional regular polytope onto a random line? The next two theorems answer this question. The proofs are postponed to Section 6.

Theorem 3.6.

The random width of the cube satisfies the following central limit theorem:

After the acceptance of this paper we became aware of the reference [18] where the central limit theorem was established for the volume of the projection of the cube onto a random linear subspace of any fixed dimension.

Theorem 3.7.

For the random width of the simplex and the crosspolytope we have

(26)
(27)

where are independent random variables with the Gumbel double exponential distribution , .

Remark 3.8.

It is easy to check that the density of equals , , where

is the modified Bessel function of the second kind.

4. Proof of Theorem 2.1

As already mentioned, the first estimate in Theorem 2.1 is a consequence of the Slepian lemma. Therefore, we concentrate on proving the inequality

For the inequality follows by direct calculations, thus we assume that .

The idea of the proof of goes back to the work of Chatterjee (see [6] or [1, p. 50]). For consider a centered Gaussian vector

with correlations defined by

and otherwise. We have

Hence it is sufficient to show that the function

is non-increasing on . Consider the function

It is immediate that

Therefore we only need to show that for any the function

is non-increasing on .

In what follows, stands for . Set and let us denote by the probability density function of . It is a well-known property of that

Therefore,

We have

and otherwise. Thus we obtain

It is easy to check that