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Technische Universität Dresden

Herausgeber: Der Rektor

Permanence of moment estimates for p-products of convex bodies

Ulrich Brehm^{1}, Hendrik Vogt^{2,3}, and Jürgen Voigt^{2,4}

1Institut für Geometrie ^{2}Institut für Analysis MATH-GE-02-00 MATH-AN-01-00

Permanence of moment estimates for p-products of convex bodies

Ulrich Brehm^{1}, Hendrik Vogt^{2,3}, and Jürgen Voigt^{2,4}

Abstract. It is shown that two inequalities concerning second and fourth moments of isotropic normed convex bodies in ℝ^{n} are permanent under forming p-products. These inequalities are connected with a concentration of mass property as well as with a central limit property. An essential tool are certain monotonicity properties of the Γ-function.

MSC 2000: 52A20 (primary), 33B15, 60F25 (secondary)

Keywords: convex body, isotropic, moment inequalities, Gamma function, central limit theorem

Introduction

The topic of the present paper originated from and is connected with a version of the central limit theorem in the context of convex bodies in geometry.

A normed convex body K ⊆ ℝ^{n} is a convex compact set of volume 1 whose centre of mass is at the origin. A normed convex body is isotropic if its ellipsoid of inertia is a Euclidean ball, i.e., if

L_{K}^{2} := ∫_{K}(x · u)^{2} dx

is independent of u ∈ S^{n−1}, the unit sphere of ℝ^{n}.

The main objective of the present paper is the validity of two inequalities concerning the moments

M_{2,n}(K) := ∫_{K} |x|^{2} dx,
M_{4,n}(K) := ∫_{K} |x|^{4} dx

of isotropic normed convex bodies K ⊆ ℝ^{n}. Here and in the following, the norm | · | always denotes the Euclidean norm. In [9] it has been shown that the inequalities

M_{2,n}(K) ⩽ M_{2,n}(Δ_{n}) = \frac{n}{(n + 1)(n + 2)} \frac{n!^{\frac{2}{n}}}{(n + 1)^{\frac{1}{n}}} ,  (0.1)

\frac{M_{4,n}(K)}{M_{2,n}(K)^{2}} ⩽ \frac{M_{4,n}(Δ_{n})}{M_{2,n}(Δ_{n})^{2}} = \frac{(n + 1)(n + 2)}{(n + 3)(n + 4)}(1 + \frac{2}{n} + \frac{6}{n + 1})  (0.2)

Typeset by L^{A}TEX 2ε

3supported by the Deutsche Forschungsgemeinschaft 4partly supported by the Deutsche Forschungsgemeinschaft Address: Technische Universität Dresden, Fachrichtung Mathematik, 01062 Dresden, Germany e-mail: brehm@math.tu-dresden.de, vogt@math.tu-dresden.de, voigt@math.tu-dresden.de

Permanence of moment estimates for p-products of convex bodies 3

are valid for isotropic normed Euclidean balls, cubes, and cross polytopes, where Δ_{n} denotes the normed regular simplex in ℝ^{n}. Moreover, it is shown that these inequalities are persistent under the operations of forming isotropic normed cones, cartesian products, and joins. Here we extend these results and show that (0.1), (0.2) are also persistent under forming p-products of isotropic normed convex bodies. This implies, in particular, that (0.1), (0.2) are valid for p-balls in ℝ^{n} (ℓ_{p}^{n}-balls).

In order to motivate these investigations we recall the notion of central limit property for isotropic normed convex bodies. We regard an isotropic normed convex body K ⊆ ℝ^{n}, with the Lebesgue measure λ_{n}, as a probability space. For a unit vector u ∈ S^{n−1} we define a random variable X_{K,u}: K → ℝ by

X_{K,u}(x) := x · u.

The density of its distribution is given by

φ_{K,u}(t) = λ_{n−1}({x ∈ K; x · u = t}) (t ∈ ℝ).

The central limit property of a set of isotropic normed convex bodies roughly means that the distributions of X_{K,u} are close to Gaussian distributions for large n and ‘most’ directions u ∈ S^{n−1}.

In order to make this more precise we introduce the set K_{n} of all isotropic normed convex bodies in ℝ^{n}, and we define

K := ⋃_{n=1}^{∞} K_{n}.

Following [3], [2], we say that a set M ⊆ K has the central limit property if

sup_{K∈K_{n}∩M} μ_{n−1}({u ∈ S^{n−1}; ||φ_{K,u} − g_{L_{K}^{2}}||_{1} > ε}) → 0 as n → ∞, for all ε > 0.

Here,

g_{σ^{2}}(t) = \frac{1}{\sqrt{2πσ^{2}}} exp (−\frac{t^{2}}{2σ^{2}} )

denotes the Gaussian density, and μ_{n−1} the surface measure on S^{n−1}, normed to a probability measure. In [2] it is shown that the central limit property can be expressed equivalently using distribution functions or the L_{∞}-norm instead of the L_{1}-norm (the latter under an additional asssumption). In [3], the central limit property is shown for the set of cubes and Euclidean balls. In [1] it is shown that, for symmetric isotropic normed convex bodies, the central limit property (in terms of distribution functions) can be derived from a ‘concentration hypothesis’. In a more general context, it is shown in [10] that a concentration of mass property implies a central limit property. This concentration of mass property (or concentration hypothesis) means that, for high dimensions, the volume of the bodies under consideration is mainly concentrated in a certain spherical shell. Using [10], and assuming the concentration of mass property for a set M ⊆ K, it is shown in [2] that M satisfies the central limit property formulated above.

The role of inequalities (0.1), (0.2) in this context is that they imply the concentration of mass property; cf. [9]. As a consequence, the subset T of all bodies K in K satisfying these inequalities satisfies the central limit property. An inequality similar to (but weaker than) (0.2) for ℓ_{p}^{n}-balls was shown in [1] in order to conclude the concentration hypothesis in this case. The desire to prove (0.1), (0.2) for ℓ_{p}^{n}-balls was the starting point of the present paper. We refer to [4] for a study of the asymptotic behaviour, as n → ∞, of the volume of sections of ℓ_{p}^{n}-balls (0 < p ⩽ ∞).

As a side remark we mention that for bodies K ∈ K satisfying (0.1) one obtains L_{K} ⩽ \frac{1}{e}. It is an open question whether there is a universal bound on L_{K} for K ∈ K (see [6; Sec. 5]). As a second remark we mention that the bound in (0.2) converges to 1 as n → ∞, which is the essential feature for deriving the concentration of mass property (cf. [9; Sec. 1]).

In Section 1 we define the (isotropic normed) p-product of isotropic normed convex bodies, and we compute the moments in terms of the moments of the given bodies.

In Section 2 we show that inequalities (0.1) and (0.2) are permanent under forming p-products.

In Section 3 we give more precise information on the p-balls in ℝ^{n}.

The most sophisticated material of this paper, in our opinion, is contained in Section 4. Having derived the expressions (1.5) and (1.8), it was not obvious how to handle the abundance of Γ-functions occurring in these expressions. The technical tools concerning the Γ-function presented in this section are essential for Sections 2 and 3 and should also be of independent interest.

Finally, in Section 5 we show that the expression \frac{M_{4,n}(K)}{M_{2,n}(K)^{2}} considered in (0.2) takes its minimal value for the Euclidean ball. The corresponding property for M_{2,n}(K) is immediate.

1 The p-product of convex bodies and its moments

For j = 1, 2, we assume n_{j} ∈ ℕ, and we take an isotropic normed convex body K_{j} ⊆ ℝ^{n_{j}}. Let | · |_{j} denote the gauge functional of K_{j}. (In general, | · |_{j} will be sublinear; it will be a norm if and only if K_{j} is symmetric.) Moreover, define n := n_{1} + n_{2}, and let 1 ⩽ p ⩽ ∞. On ℝ^{n} we define the sublinear functional

|(x_{1}, x_{2})|_{p} := \{ \begin{matrix}(|x_{1}|_{1}^{p} + |x_{2}|_{2}^{p})^{1/p} & for 1 ⩽ p < ∞, \\ max(|x_{1}|_{1}, |x_{2}|_{2}) & for p = ∞, \end{matrix}

where x_{j} ∈ ℝ^{n_{j}} (j = 1, 2). We define the p-product K_{1} ×_{p} K_{2} (which is not a normed convex body) by

K_{1} ×_{p} K_{2} := {(x_{1}, x_{2}) ∈ ℝ^{n}; |(x_{1}, x_{2})|_{p} ⩽ 1}.

We note that K_{1} ×_{∞} K_{2} is just the ordinary cartesian product, whereas

K_{1} ×_{1} K_{2} = {(tx_{1}, (1 − t)x_{2}); x_{j} ∈ K_{j} (j = 1, 2), 0 ⩽ t ⩽ 1}

is the join of K_{1} and K_{2}. These cases were studied in [9].

Examples of p-products

p = 1  p = \frac{4}{3}  p = 2  p = 4  p = ∞

Figure 1

Note that ∫_{K_{1}×_{p}K_{2}}(x · u)(x · v) dx = 0 if u ∈ ℝ^{n_{1}} × {0}, v ∈ {0} × ℝ^{n_{2}}, because the centre of mass of K_{1} and K_{2}, respectively, is the origin. Moreover, ∫_{K_{1}×_{p}K_{2}}(x · u)^{2} dx is constant on S^{n_{1}−1} × {0} and on {0} × S^{n_{2}−1} because K_{1} and K_{2} are isotropic. Therefore, the isotropic normed body associated with K_{1} ×_{p} K_{2} is of the form (α_{1}K_{1}) ×_{p} (α_{2}K_{2}), where α_{1}, α_{2} are suitable scaling factors. In Figure 1 we show some examples for the case n_{1} = 2, n_{2} = 1, with different p, where K_{1} is a circle, a square and an equilateral triangle, respectively (and K_{2} = [−\frac{1}{2}, \frac{1}{2}]).

In order to find the correct scaling factors we first compute the volume of K_{1} ×_{p} K_{2}. The method we use for doing so is inspired by the computation of the volume of p-balls given in [7; p.11].

First we note

∫_{ℝ^{n}} e^{−|(x_{1},x_{2})|_{p}^{p}} dx = ∫_{ℝ^{n_{1}}} e^{−|x_{1}|_{1}^{p}} dx_{1} ∫_{ℝ^{n_{2}}} e^{−|x_{2}|_{2}^{p}} dx_{2}.  (1.1)

Now,

∫_{ℝ^{n_{1}}} e^{−|x_{1}|_{1}^{p}} dx_{1} = ∫_{ℝ^{n_{1}}} (− ∫_{|x_{1}|_{1}^{p}}^{∞} \frac{d}{dt}e^{−t} dt) dx_{1}
= ∫_{0}^{∞} ∫_{|x_{1}|_{1}^{p}⩽t} dx_{1} e^{−t} dt
= ∫_{0}^{∞} t^{\frac{n_{1}}{p}} e^{−t} dt V_{1} = Γ(\frac{n_{1}}{p} + 1),

with V_{1} := vol_{n_{1}}(K_{1}) = 1. Computing the other integrals in (1.1) in the same way, we obtain

Γ(\frac{n}{p} + 1) vol_{n}((α_{1}K_{1}) ×_{p} (α_{2}K_{2})) = α_{1}^{n_{1}}Γ(\frac{n_{1}}{p} + 1)α_{2}^{n_{2}}Γ(\frac{n_{2}}{p} + 1),

and therefore the first requirement for the scaling factors α_{1}, α_{2} > 0 amounts to

α_{1}^{n_{1}}α_{2}^{n_{2}} = \frac{Γ(\frac{n}{p} + 1)}{Γ(\frac{n_{1}}{p} + 1)Γ(\frac{n_{2}}{p} + 1)}.  (1.2)

In order to obtain isotropy we compute second moments. (Recall that | · | denotes the Euclidean norm.)

∫_{ℝ^{n}} |x_{1}|^{2}e^{−|(x_{1},x_{2})|_{p}^{p}} dx = ∫_{ℝ^{n_{1}}} |x_{1}|^{2} (− ∫_{|x_{1}|_{1}^{p}}^{∞} \frac{d}{dt}e^{−t} dt) dx_{1} ∫_{ℝ^{n_{2}}} e^{−|x_{2}|_{2}^{p}} dx_{2}
= ∫_{0}^{∞} ∫_{|x_{1}|_{1}^{p}⩽t} |x_{1}|^{2} dx_{1} e^{−t} dt Γ(\frac{n_{2}}{p} + 1)
= ∫_{0}^{∞} t^{\frac{n_{1}+2}{p}} M_{2,1}e^{−t} dt Γ(\frac{n_{2}}{p} + 1)
= Γ(\frac{n_{1}+2}{p} + 1)Γ(\frac{n_{2}}{p} + 1)M_{2,1},

where, for abbreviation, M_{2,j} := M_{2,n_{j}}(K_{j}) (j = 1, 2). The first expression in the last chain of equalities can also be computed as

∫ |x_{1}|^{2}e^{−|(x_{1},x_{2})|_{p}^{p}} dx = ∫ |x_{1}|^{2} (− ∫_{|x|_{p}^{p}}^{∞} \frac{d}{dt}e^{−t} dt) dx
= ∫_{0}^{∞} ∫_{|x|_{p}^{p}⩽t} |x_{1}|^{2} dx e^{−t} dt
= ∫_{0}^{∞} t^{\frac{n+2}{p}} e^{−t} dt ∫_{K_{1}×_{p}K_{2}} |x_{1}|^{2} dx
= Γ(\frac{n+2}{p} + 1) ∫_{K_{1}×_{p}K_{2}} |x_{1}|^{2} dx,

and therefore

∫_{(α_{1}K_{1})×_{p}(α_{2}K_{2})} |x_{1}|^{2} dx = α_{1}^{n_{1}+2}α_{2}^{n_{2}} \frac{Γ(\frac{n_{1}+2}{p} + 1)Γ(\frac{n_{2}}{p} + 1)}{Γ(\frac{n+2}{p} + 1)}M_{2,1}.  (1.3)

Now, isotropy requires

\frac{1}{n_{1}} ∫_{(α_{1}K_{1})×_{p}(α_{2}K_{2})} |x_{1}|^{2} dx = \frac{1}{n_{2}} ∫_{(α_{1}K_{1})×_{p}(α_{2}K_{2})} |x_{2}|^{2} dx,

which can be expressed as

α_{1}^{2}Γ(\frac{n_{1}+2}{p} + 1)Γ(\frac{n_{2}}{p} + 1)\frac{M_{2,1}}{n_{1}} = α_{2}^{2}Γ(\frac{n_{1}}{p} + 1)Γ(\frac{n_{2}+2}{p} + 1)\frac{M_{2,2}}{n_{2}} .  (1.4)

With α_{1}, α_{2} obtained from (1.2), (1.4) we define

K_{1} ∗_{p} K_{2} := (α_{1}K_{1}) ×_{p} (α_{2}K_{2})

Permanence of moment estimates for p-products of convex bodies 7

as the isotropic normed p-product of K_{1} and K_{2}; we obtain

M_{2,n}(K_{1} ∗_{p} K_{2}) = n (\frac{1}{n_{1}} ∫_{K_{1}∗_{p}K_{2}} |x_{1}|^{2} dx)^{\frac{n_{1}}{n}} (\frac{1}{n_{2}} ∫_{K_{1}∗_{p}K_{2}} |x_{2}|^{2} dx)^{\frac{n_{2}}{n}}

= n (α_{1}^{n_{1}+2}α_{2}^{n_{2}} \frac{Γ(\frac{n_{1}+2}{p} + 1)Γ(\frac{n_{2}}{p} + 1)}{Γ(\frac{n+2}{p} + 1)} \frac{M_{2,1}}{n_{1}} )^{\frac{n_{1}}{n}} ×

(α_{1}^{n_{1}}α_{2}^{n_{2}+2} \frac{Γ(\frac{n_{1}}{p} + 1)Γ(\frac{n_{2}+2}{p} + 1)}{Γ(\frac{n+2}{p} + 1)} \frac{M_{2,2}}{n_{2}} )^{\frac{n_{2}}{n}}

= n [\frac{(α_{1}^{n_{1}}α_{2}^{n_{2}})^{n+2}}{Γ(\frac{n+2}{p} + 1)^{n}} (Γ(\frac{n_{1}+2}{p} + 1)Γ(\frac{n_{2}}{p} + 1)\frac{M_{2,1}}{n_{1}} )^{n_{1}} ×

(Γ(\frac{n_{1}}{p} + 1)Γ(\frac{n_{2}+2}{p} + 1)\frac{M_{2,2}}{n_{2}} )^{n_{2}}]^{\frac{1}{n}} .

By (1.2), this can be transformed into

M_{2,n}(K_{1} ∗_{p} K_{2}) = n (\frac{f(n_{1}, \frac{1}{p})f(n_{2}, \frac{1}{p})}{f(n, \frac{1}{p})} (\frac{M_{2,n_{1}}(K_{1})}{n_{1}})^{n_{1}} (\frac{M_{2,n_{2}}(K_{2})}{n_{2}})^{n_{2}})^{\frac{1}{n}} ,  (1.5)

with the function

f(n, α) := \frac{Γ(α(n + 2) + 1)^{n}}{Γ(αn + 1)^{n+2}} .  (1.6)

The derivation of this formula is only valid for p < ∞, but the final expression is also true for p = ∞ (cartesian product); cf. [9; Sec. 3].

In order to compute \frac{M_{4,n}}{M_{2,n}^{2}} (K_{1} ∗_{p} K_{2}) we note

∫_{K_{1}×_{p}K_{2}} |x|^{4} dx = ∫_{K_{1}×_{p}K_{2}}(|x_{1}|^{4} + 2|x_{1}|^{2}|x_{2}|^{2} + |x_{2}|^{4}) dx.

In the same way as above we compute

∫_{K_{1}×_{p}K_{2}} |x_{1}|^{4} dx = \frac{Γ(\frac{n_{1}+4}{p} + 1)Γ(\frac{n_{2}}{p} + 1)}{Γ(\frac{n+4}{p} + 1)}M_{4,1},
∫_{K_{1}×_{p}K_{2}} |x_{1}|^{2}|x_{2}|^{2} dx = \frac{Γ(\frac{n_{1}+2}{p} + 1)Γ(\frac{n_{2}+2}{p} + 1)}{Γ(\frac{n+4}{p} + 1)}M_{2,1}M_{2,2}.

In the following computations we use the fact that M_{2} := M_{2,n}(K_{1} ∗_{p} K_{2}) can be expressed by means of (1.3) as well as the symmetric expression. Thus,

\frac{∫_{K_{1}∗_{p}K_{2}} |x_{1}|^{4} dx}{M_{2}^{2}} = \frac{α_{1}^{n_{1}+4}α_{2}^{n_{2}} \frac{Γ( \frac{n_{1}+4}{p} +1)Γ( \frac{n_{2}}{p} +1)}{Γ( \frac{n+4}{p} +1)}M_{4,1}}{\frac{n^{2}}{n_{1}^{2}} α_{1}^{2(n_{1}+2)}α_{2}^{2n_{2}} \frac{Γ( \frac{n_{1}+2}{p} +1)^{2}Γ( \frac{n_{2}}{p} +1)^{2}}{Γ( \frac{n+2}{p} +1)^{2}} M_{2,1}^{2}}

= \frac{n_{1}^{2}}{n^{2}} \frac{1}{α_{1}^{n_{1}}α_{2}^{n_{2}}} \frac{Γ(\frac{n+2}{p} + 1)^{2}}{Γ(\frac{n+4}{p} + 1)} \frac{Γ(\frac{n_{1}+4}{p} + 1)}{Γ(\frac{n_{1}+2}{p} + 1)^{2}} \frac{1}{Γ(\frac{n_{2}}{p} + 1)} \frac{M_{4,1}}{M_{2,1}^{2}}

= \frac{n_{1}^{2}}{n^{2}} \frac{g(n_{1}, \frac{1}{p})}{g(n, \frac{1}{p})} \frac{M_{4,1}}{M_{2,1}^{2}} ,

with the function

g(n, α) := \frac{Γ(αn + 1)Γ(α(n + 4) + 1)}{Γ(α(n + 2) + 1)^{2}},  (1.7)

and

\frac{∫_{K_{1}∗_{p}K_{2}} |x_{1}|^{2}|x_{2}|^{2} dx}{M_{2}^{2}}

= \frac{α_{1}^{n_{1}+2}α_{2}^{n_{2}+2} \frac{Γ( \frac{n_{1}+2}{p} +1)Γ( \frac{n_{2}+2}{p} +1)}{Γ( \frac{n+4}{p} +1)}M_{2,1}M_{2,2}}{\frac{n}{n_{1}} α_{1}^{n_{1}+2}α_{2}^{n_{2}} \frac{Γ( \frac{n_{1}+2}{p} +1)Γ( \frac{n_{2}}{p} +1)}{Γ( \frac{n+2}{p} +1)}M_{2,1} \frac{n}{n_{2}} α_{1}^{n_{1}}α_{2}^{n_{2}+2} \frac{Γ( \frac{n_{1}}{p} +1)Γ( \frac{n_{2}+2}{p} +1)}{Γ( \frac{n+2}{p} +1)}M_{2,2}}

= \frac{n_{1}n_{2}}{n^{2}} \frac{1}{α_{1}^{n_{1}}α_{2}^{n_{2}}} \frac{Γ(\frac{n+2}{p} + 1)^{2}}{Γ(\frac{n+4}{p} + 1)} \frac{1}{Γ(\frac{n_{1}}{p} + 1)Γ(\frac{n_{2}}{p} + 1)}

= \frac{n_{1}n_{2}}{n^{2}} \frac{1}{g(n, \frac{1}{p})}.

Summing up, we obtain

\begin{aligned}\frac{M_{4,n}}{M_{2,n}^{2}} & (K_{1} ∗_{p} K_{2}) \\ & = \frac{1}{n^{2}g(n, \frac{1}{p})} (n_{1}^{2}g(n_{1}, \frac{1}{p})\frac{M_{4,n_{1}}}{M_{2,n_{1}}^{2}} (K_{1}) + 2n_{1}n_{2} + n_{2}^{2}g(n_{2}, \frac{1}{p})\frac{M_{4,n_{2}}}{M_{2,n_{2}}^{2}} (K_{2})) . \end{aligned}  (1.8)

2 Permanence of inequalities for the second and fourth moments

We assume n_{j}, K_{j} given as in Section 1, for j = 1, 2.

Theorem 2.1. The function

p ⟼ M_{2,n}(K_{1} ∗_{p} K_{2})

is strictly decreasing on [1, 2], strictly increasing on [2, ∞]; in particular, it attains its minimum for p = 2. Moreover, M_{2,n}(K_{1} ∗_{p} K_{2}) attains its maximum for p = ∞. If K_{1} and K_{2} satisfy (0.1), then K_{1} ∗_{p} K_{2} also satisfies (0.1) for all p ∈ [1, ∞].

Proof. The function \frac{f(n_{1}, \frac{1}{p})f(n_{2}, \frac{1}{p})}{f(n, \frac{1}{p})} occurring in the expression for M_{2,n}(K_{1} ∗_{p} K_{2}), see (1.5), is treated by computing the logarithmic derivative

∂_{α} ln \frac{f(n_{1}, α)f(n_{2}, α)}{f(n, α)} = ∂_{α} ln f(n, α) (\frac{∂_{α} ln f(n_{1}, α)}{∂_{α} ln f(n, α)} + \frac{∂_{α} ln f(n_{2}, α)}{∂_{α} ln f(n, α)} − 1).

By Corollary 4.3, the second factor is strictly increasing and vanishes at α = \frac{1}{2}, whereas the first factor is positive. This implies the first assertion.

It is also clear that M_{2,n}(K_{1} ∗_{p} K_{2}) attains its maximum for p = 1 or p = ∞. In order to exclude p = 1 it is therefore sufficient to show

\frac{f(n_{1}, 1)f(n_{2}, 1)}{f(n, 1)} ⩽ 1 (= \frac{f(n_{1}, 0)f(n_{2}, 0)}{f(n, 0)}) .

This is shown subsequently in Lemma 2.2.

In order to prove the last statement it is therefore sufficient to check (0.1) for p = ∞, i.e., the isotropic normed cartesian product. This case, however, has been treated in [9].

Lemma 2.2. (a) The function

k ⟼ \frac{f(k − 1, 1)}{f(k, 1)}

(where f(k, 1) = \frac{Γ((k+2)+1)^{k}}{Γ(k+1)^{k+2}} = \frac{(k+2)!^{k}}{k!^{k+2}} = \frac{((k+1)(k+2))^{k}}{k!^{2}} , see (1.6)) is monotone decreasing in k ∈ ℕ.

(b) If n, n_{1}, n_{2} ∈ ℕ_{0}, n = n_{1} + n_{2}, then

\frac{f(n_{1}, 1)f(n_{2}, 1)}{f(n, 1)} ⩽ 1.

Proof. (a) An elementary computation yields

\frac{f(k − 1, 1)}{f(k, 1)} = \frac{k}{k + 1} (\frac{k}{k + 2})^{k} = \frac{k}{k + 1} (1 − \frac{2}{k + 2})^{k} .

For k = 1, 2 we obtain

\frac{f(0, 1)}{f(1, 1)} = \frac{f(1, 1)}{f(2, 1)} = \frac{1}{6}.

For the logarithmic derivative we compute and estimate

\frac{d}{dk} ln (\frac{k}{k + 1} (1 − \frac{2}{k + 2})^{k}) = \frac{1}{k} − \frac{1}{k + 1} + ln (1 − \frac{2}{k + 2}) + \frac{k}{k} − \frac{k}{k + 2}

⩽ \frac{1}{k} − \frac{1}{k + 1} + \frac{2}{k + 2} − (\frac{2}{k + 2} + \frac{2}{(k + 2)^{2}} + \frac{8}{3(k + 2)^{3}} )

= \frac{−k^{3} − \frac{8}{3}k^{2} + \frac{16}{3}k + 8}{k(k + 1)(k + 2)^{3}},

and this is ⩽ 0 for k ⩾ 2.

(b) For arbitrary n ∈ ℕ_{0} we obviously have \frac{f(0,1)f(n,1)}{f(n,1)} = 1. If n_{1}, n_{2} ∈ ℕ_{0}, n_{1} < n_{2}, n_{1} + n_{2} = n, then (a) implies

\frac{f(n_{1} + 1, 1)f(n_{2} − 1, 1)}{f(n, 1)} ⩽ \frac{f(n_{1}, 1)f(n_{2}, 1)}{f(n, 1)}

which by induction is ⩽ 1.

Theorem 2.3. There exists p_{0} ∈ [2, ∞) such that

p ⟼ \frac{M_{4,n}}{M_{2,n}^{2}} (K_{1} ∗_{p} K_{2})

is strictly decreasing on [1, p_{0}], strictly increasing on [p_{0}, ∞]. Here p_{0} = 2 if and only if K_{j} = B_{n_{j},2}, the normed Euclidean ball in ℝ^{n_{j}}, for j = 1, 2. In particular, \frac{M_{4,n}}{M_{2,n}^{2}} (K_{1} ∗_{p} K_{2}) attains its maximum for p = 1 or p = ∞. If K_{1} and K_{2} satisfy (0.2), then K_{1} ∗_{p} K_{2} also satisfies (0.2) for all p ∈ [1, ∞].

As a preparation for the proof, we treat the expression occurring in (1.8) in a more general setting.

Lemma 2.4. Let n, n_{1}, n_{2} > 0, n > n_{1}, n_{2}, c_{1}, d > 0, c_{2} ⩾ 0, and define

G(α) := \frac{1}{g(n, α)}(c_{1}g(n_{1}, α) + d + c_{2}g(n_{2}, α))

for α ⩾ 0. Then there exists α_{0} > 0 such that G is strictly decreasing on [0, α_{0}], strictly increasing on [α_{0}, ∞). In particular, G attains its maximum on [0, 1] at α = 0 or at α = 1.

Proof. We compute

G′(α) = \frac{1}{g(n, α)}(c_{1}∂_{α}g(n_{1}, α) + c_{2}∂_{α}g(n_{2}, α))

− \frac{∂_{α}g(n, α)}{g(n, α)^{2}}(c_{1}g(n_{1}, α) + d + c_{2}g(n_{2}, α))

= \frac{∂_{α} ln g(n, α)}{g(n, α)} [c_{1} (\frac{∂_{α} ln g(n_{1}, α)}{∂_{α} ln g(n, α)} − 1) g(n_{1}, α) − d

+c_{2} (\frac{∂_{α} ln g(n_{2}, α)}{∂_{α} ln g(n, α)} − 1) g(n_{2}, α)] .

(2.1)

By Corollary 4.3, the second factor is strictly increasing (note that g(n, α) is increasing, by the positivity of ∂_{α} ln g(n, α)) and negative for small α since d > 0. Since the first factor is positive it follows that G′(α) has a unique zero α_{0}. (It is not difficult to show that, as α → ∞, the second factor tends to infinity because g(n_{1}, α) → ∞.)

Proof of Theorem 2.3. Because of formula (1.8), the first statement follows from Lemma 2.4, except for the fact that p_{0} ⩾ 2. If K_{j} = B_{n_{j},2} for j = 1, 2, then Theorem 5.1 implies p_{0} = 2; in particular, in Lemma 2.4, with c_{j} = n_{j}^{2} \frac{M_{4,n_{j}}}{M_{2,n_{j}}^{2}} (B_{n_{j},2}) (j = 1, 2), d = 2n_{1}n_{2},

one has G′(\frac{1}{2}) = 0. Now, Theorem 5.1 implies G′(\frac{1}{2}) > 0 in all other cases (cf. formula (2.1)), i.e., p_{0} > 2.

In order to prove the last statement it is therefore sufficient to check (0.2) for p = 1 and p = ∞, i.e., the isotropic normed join and the isotropic normed cartesian product. These cases have been treated in [9].

3 Second and fourth moments for the p-balls

For n ∈ ℕ, p ∈ [1, ∞], we denote by B_{n,p} the normed ball in ℝ^{n} corresponding to the p-norm | · |_{p}. The volume of the unit p-ball

B̌_{n,p} := {x ∈ ℝ^{n}; |x|_{p} ⩽ 1}

is given by

vol_{n}(B̌_{n,p}) = \frac{(2Γ(\frac{1}{p} + 1))^{n}}{Γ(\frac{n}{p} + 1)}

(cf. [7; p.11]); therefore

B_{n,p} = \frac{Γ(\frac{n}{p} + 1)^{\frac{1}{n}}}{2Γ(\frac{1}{p} + 1)} B̌_{n,p}.

For n_{1}, n_{2} ∈ ℕ, n = n_{1} + n_{2}, one obviously obtains

B_{n,p} = B_{n_{1},p} ∗_{p} B_{n_{2},p},

and therefore recursion formulas for M_{2,n}(B_{n,p}) and \frac{M_{4,n}}{M_{2,n}^{2}} (B_{n,p}) are obtained from (1.5) and (1.8).

Taking into account

M_{2,1}(B_{1,p}) = \frac{1}{12}, \frac{M_{4,1}}{M_{2,1}^{2}} (B_{1,p}) = \frac{9}{5}

one obtains

M_{2,n}(B_{n,p}) = \frac{n}{12} \frac{f(1, \frac{1}{p})}{f(n, \frac{1}{p})^{\frac{1}{n}}} ,

\frac{M_{4,n}}{M_{2,n}^{2}} (B_{n,p}) = \frac{1}{g(n, \frac{1}{p})} (\frac{9}{5n}g(1, \frac{1}{p}) + 1 − \frac{1}{n}) .

Theorem 3.1. The functions

p ⟼ M_{2,n}(B_{n,p}), p ⟼ \frac{M_{4,n}}{M_{2,n}^{2}} (B_{n,p})

are strictly decreasing on [1, 2], strictly increasing on [2, ∞].

Proof. For the first function we compute the logarithmic derivative

∂_{α} ln \frac{f(1, α)}{f(n, α)^{\frac{1}{n}}} = ∂_{α} ln f(n, α) (\frac{∂_{α} ln f(1, α)}{∂_{α} ln f(n, α)} − \frac{1}{n}) .

By Corollary 4.3, this implies the assertion for the first function as in the proof of Theorem 2.1.

For the second function, we apply Lemma 2.4 (with c_{2} = 0); note that α_{0} = \frac{1}{2} follows from Theorem 5.1 (or can be obtained by direct computation).

4 Monotonicity properties of the Γ-function

In this section we study properties of functions of the type

F (α) := ∏_{j=1}^{N} Γ(αn_{j} + 1)^{γ_{j}} (α ⩾ 0),

where n_{j} ⩾ 0, γ_{j} ∈ ℝ (j = 1, …, N). We will make use of the following integral representation of the logarithmic derivative of the Γ-function,

Ψ(x) := \frac{d}{dx} ln Γ(x + 1) = ∫_{0}^{∞} (\frac{e^{−t}}{t} − \frac{e^{−xt}}{e^{t} − 1}) dt

(cf. [5; §1, formula (60)] or [8; 6.351]). If ∑_{j=1}^{N} γ_{j}n_{j} = 0 then the logarithmic derivative of F can be expressed as

\begin{aligned}\frac{d}{dα} ln F (α) & = ∑_{j} γ_{j}n_{j}Ψ(αn_{j}) \\ & = − ∫_{0}^{∞} \frac{1}{e^{t} − 1} ∑_{j} γ_{j}n_{j}e^{−αn_{j}t}dt \\ & = \frac{1}{4α} ∫_{0}^{∞} \frac{1}{sinh^{2}(\frac{t}{2})} ∑_{j} γ_{j}(e^{−αn_{j}t} − 1)dt, \end{aligned}  (4.1)

where in the last step we used partial integration and \frac{d}{dt}(\frac{1}{e^{t}−1}) = −\frac{e^{t}}{(e^{t}−1)^{2}} = −\frac{1}{4 sinh^{2}( \frac{t}{2} )}. Note that the functions f and g needed in Sections 2 and 3 are of the above type, with ∑_{j} γ_{j}n_{j} = 0.

Proposition 4.1. Let k_{j} ⩾ 0, m_{j} ∈ ℝ (j = 1, …, N) be such that the function

η(x) := ∑_{j} m_{j}x^{k_{j}}

has no zeros on (0, 1).

(a) Define φ(n, α) := ∏_{j} Γ(α(n + k_{j}) + 1)^{\frac{m_{j}}{n+k_{j}}} (n > 0, α ⩾ 0) and assume η(1) = 0. Then

∂_{α} ln φ(n, α) = −\frac{1}{α} ∫_{0}^{∞} \frac{η(e^{−t})}{e^{t/α} − 1}e^{−nt} dt (α > 0)

and, for n > n_{1} > 0, the function

Q(α) := \frac{∂_{α} ln φ(n_{1}, α)}{∂_{α} ln φ(n, α)}

is strictly increasing on (0, ∞). If, in addition, η′(1) ≠ 0 then Q(α) → 1 as α → 0.

(b) Define ψ(n, α) := ∏_{j} Γ(α(n + k_{j}) + 1)^{m_{j}} (n, α ⩾ 0) and assume η(1) = η′(1) = 0. Then

∂_{α} ln ψ(n, α) = \frac{1}{4α^{2}} ∫_{0}^{∞} \frac{η(e^{−t})}{sinh^{2}(\frac{t}{2α})}e^{−nt} dt (α > 0)

and, for n > n_{1} ⩾ 0, the function

Q(α) := \frac{∂_{α} ln ψ(n_{1}, α)}{∂_{α} ln ψ(n, α)}

is strictly increasing on (1, ∞). If, in addition, η′′(1) ≠ 0 then Q(α) → 1 as α → 0.

For the proof we need the following elementary tool.

Lemma 4.2. Let f_{1}, f_{2}, h: (0, ∞) → (0, ∞) be continuous functions, \frac{f_{2}}{f_{1}} strictly increasing, h strictly increasing. Then

\frac{∫_{0}^{∞} f_{1}(t)h(t) dt}{∫_{0}^{∞} f_{1}(t) dt} < \frac{∫_{0}^{∞} f_{2}(t)h(t) dt}{∫_{0}^{∞} f_{2}(t) dt}.

(All the integrals are assumed to exist.)

Proof. Without restriction ∫_{0}^{∞} f_{1}(t) dt = ∫_{0}^{∞} f_{2}(t) dt = 1. Then ∫_{0}^{∞}(\frac{f_{2}}{f_{1}} − 1)f_{1} dt = 0, so there exists τ > 0 such that \frac{f_{2}(τ)}{f_{1}(τ)} = 1. We conclude

∫_{0}^{∞}(f_{2} − f_{1})h dt = ∫_{0}^{τ} (\frac{f_{2}}{f_{1}} − 1) f_{1}h dt + ∫_{τ}^{∞} (\frac{f_{2}}{f_{1}} − 1) f_{1}h dt

> h(τ) ∫_{0}^{∞} (\frac{f_{2}}{f_{1}} − 1) f_{1} dt = 0.

Proof of Proposition 4.1. Without restriction assume η ⩾ 0. (Otherwise replace m_{j} by −m_{j} for j = 1, …, N.)

(a) Since ∑_{j} \frac{m_{j}}{n+k_{j}} (n + k_{j}) = η(1) = 0 we obtain from (4.1) the formula

∂_{α} ln φ(n, α) = − ∫_{0}^{∞} \frac{1}{e^{t} − 1}e^{−αnt}η(e^{−αt}) dt = −\frac{1}{α} ∫_{0}^{∞} \frac{η(e^{−t})}{e^{t/α} − 1}e^{−nt} dt.  (4.2)

Letting f_{α}(t) := \frac{η(e^{−t})}{e^{t/α}−1}e^{−nt} and h(t) := e^{(n−n_{1})t} we get

Q(α) = \frac{∫_{0}^{∞} f_{α}(t)h(t) dt}{∫_{0}^{∞} f_{α}(t) dt}.

By Lemma 4.2, for the proof of the strict monotonicity of Q it remains to show that, for 0 < α_{1} < α_{2}, the function

\frac{f_{α_{2}}(t)}{f_{α_{1}}(t)} = \frac{e^{t/α_{1}} − 1}{e^{t/α_{2}} − 1}

is strictly increasing (on (0, ∞)), or equivalently, that s ↦ \frac{e^{ps}−1}{e^{s}−1}, where p > 1, is strictly increasing. The numerator of the derivative of this function,

\frac{p − 1}{p}e^{(p+1)s} + \frac{1}{p}e^{s} − e^{ps},

is positive by the convexity of the exponential function.

For the proof of Q(α) → 1 (α → 0) we note that

∂_{α} ln φ(n, 0) = 0, ∂_{α}^{2} ln φ(n, 0) = ∫_{0}^{∞} \frac{t}{e^{t} − 1}η′(1) dt

by (4.2) and the assumption η(1) = 0, so we are done by de l’Hospital’s rule.

(b) Since ∑_{j} m_{j} = η(1) = 0 and ∑_{j} m_{j}(n + k_{j}) = nη(1) + η′(1) = 0 we obtain from (4.1) the formula

∂_{α} ln ψ(n, α) = \frac{1}{4α} ∫_{0}^{∞} \frac{1}{sinh^{2}(\frac{t}{2})}e^{−αnt}η(e^{−αt}) dt = \frac{1}{4α^{2}} ∫_{0}^{∞} \frac{η(e^{−t})}{sinh^{2}(\frac{t}{2α})}e^{−nt} dt. (4.3)

Letting f_{α}(t) := \frac{η(e^{−t})}{sinh^{2}( \frac{t}{2α})}e^{−nt} and h(t) := e^{(n−n_{1})t}, we get

Q(α) = \frac{∫_{0}^{∞} f_{α}(t)h(t) dt}{∫_{0}^{∞} f_{α}(t) dt}.

Again by Lemma 4.2, for the proof of the strict monotonicity of Q it remains to show that, for 0 < α_{1} < α_{2}, the function

\frac{f_{α_{2}}(t)}{f_{α_{1}}(t)} = (\frac{sinh(\frac{t}{2α_{1}} )}{sinh(\frac{t}{2α_{2}} )})^{2}

is strictly increasing (on (0, ∞)), or equivalently, that s ↦ \frac{sinh(ps)}{sinh s}, where p > 1, is strictly increasing. Differentiation transforms this into the question whether

p coth(ps) > coth s.

The latter is a consequence of the strict monotonicity of

(0, ∞) ∋ p ↦ p coth p,

which in turn follows from

\frac{d}{dp}(p coth p) = \frac{cosh p sinh p − p}{(sinh p)^{2}} > \frac{1 · p − p}{(sinh p)^{2}} = 0.

For the proof of Q(α) → 1 (α → 0) we apply de l’Hospital’s rule twice to the first representation of ∂_{α} ln ψ(n, α) in (4.3):

lim_{α→0} Q(α) = lim_{α→0} \frac{α∂_{α} ln ψ(n_{1}, α)}{α∂_{α} ln ψ(n, α)} = lim_{α→0} \frac{∂_{α}^{2}(α∂_{α} ln ψ(n_{1}, α))}{∂_{α}^{2}(α∂_{α} ln ψ(n, α))} = 1

since ∂_{α}^{2}(α∂_{α} ln ψ(n, α))_{α=0} = \frac{1}{4} ∫_{0}^{∞} \frac{t^{2}}{sinh^{2}( \frac{t}{2} )}η′′(1) dt for all n ⩾ 0, by the assumption η(1) = η′(1) = 0.

In the following corollary we apply Proposition 4.1 to the functions f and g used in Sections 2 and 3.

Corollary 4.3. As in (1.6) and (1.7), let f(n, α) := \frac{Γ(α(n+2)+1)^{n}}{Γ(αn+1)^{n+2}} , g(n, α) :=

\frac{Γ(αn+1)Γ(α(n+4)+1)}{Γ(α(n+2)+1)^{2}} (n, α ⩾ 0). Then, for n > n_{1} > 0, the functions

α ⟼ \frac{∂_{α} ln f(n_{1}, α)}{∂_{α} ln f(n, α)} and α ⟼ \frac{∂_{α} ln g(n_{1}, α)}{∂_{α} ln g(n, α)}

are strictly increasing, ∂_{α} ln f and ∂_{α} ln g are positive on (0, ∞) × (0, ∞), ∂_{α} ln f(n, \frac{1}{2}) = 2n, and

\frac{∂_{α} ln g(n_{1}, α)}{∂_{α} ln g(n, α)} → 1 as α → 0.

Proof. To prove the statements about f we define φ := f^{\frac{1}{n(n+2)}} . Then φ is of the type considered in Proposition 4.1(a), with k_{1} = 2, k_{2} = 0, m_{1} = 1, m_{2} = −1. The corresponding function η is η(x) = x^{2} − 1, so it fulfils the conditions of Proposition 4.1(a). Hence

\frac{∂_{α} ln f(n_{1}, α)}{∂_{α} ln f(n, α)} = \frac{n_{1}(n_{1} + 2)}{n(n + 2)} \frac{∂_{α} ln φ(n_{1}, α)}{∂_{α} ln φ(n, α)}

is strictly increasing and

∂_{α} ln φ(n, α) = \frac{1}{α} ∫_{0}^{∞} \frac{1 − e^{−2t}}{e^{t/α} − 1} e^{−nt} dt > 0 (α > 0).

For α = \frac{1}{2} we obtain

∂_{α} ln f(n, \frac{1}{2}) = n(n + 2)∂_{α} ln φ(n, \frac{1}{2}) = 2n(n + 2) ∫_{0}^{∞} e^{−(n+2)t} dt = 2n.

The function g is of the type considered in Proposition 4.1(b), with k_{1} = 0, k_{2} = 4, k_{3} = 2, m_{1} = m_{2} = 1, m_{3} = −2. The corresponding function η is η(x) = 1+x^{4}−2x^{2} = (1−x^{2})^{2} ⩾ 0, so an application of Proposition 4.1(b) yields the first and the last statement about g. The positivity of ∂_{α} ln g follows from (4.3).

5 The minimum of \frac{M_{4,n}M_{0,n}}{M_{2,n}^{2}}

The result of this section will be formulated for the more general context of compact subsets of ℝ^{n} of non-vanishing volume which are star shaped with respect to the origin. A body K like this is described by its radial function ρ = ρ_{K}: S^{n−1} → [0, ∞),

ρ(u) := max{r ⩾ 0; ru ∈ K}.

The moments of K are then given by

M_{0,n}(K) = vol(K) = \frac{1}{n} ∫_{S^{n−1}} ρ(u)^{n} dS(u),
M_{2,n}(K) = \frac{1}{n + 2} ∫_{S^{n−1}} ρ(u)^{n+2} dS(u),
M_{4,n}(K) = \frac{1}{n + 4} ∫_{S^{n−1}} ρ(u)^{n+4} dS(u).

Theorem 5.1. Let K ⊆ ℝ^{n} be compact and star shaped with respect to the origin, with the origin contained in the interior of K. Then

\frac{M_{4,n}M_{0,n}}{M_{2,n}^{2}} (K) ⩾ \frac{M_{4,n}M_{0,n}}{M_{2,n}^{2}} (B_{n,2}),

with equality if and only if K is a multiple of the Euclidean ball B_{n,2}.

Proof. In contrast to the remainder of this paper, we shall choose the scaling of K in such a way that the second moment is normed to M_{2,n}(K) = n. By ρ_{0} we denote the radius of the accordingly scaled Euclidean ball,

ρ_{0} = (\frac{n(n + 2)}{σ_{n−1}})^{\frac{1}{n+2}} = (\frac{n + 2}{ω_{n}})^{\frac{1}{n+2}} .

Denoting by μ_{n−1} the surface measure on S^{n−1}, normed to a probability measure, we then have to show

∫ ρ_{0}^{n+4} dμ_{n−1} ∫ ρ_{0}^{n} dμ_{n−1} ⩽ ∫ ρ(u)^{n+4} dμ_{n−1} ∫ ρ(u)^{n} dμ_{n−1},

or equivalently,

1 ⩽ ∫ ((\frac{ρ}{ρ_{0}} )^{n+2})^{\frac{n+4}{n+2}} dμ_{n−1} ∫ ((\frac{ρ}{ρ_{0}} )^{n+2})^{\frac{n}{n+2}} dμ_{n−1},

with equality if and only if ρ = ρ_{0}. Note that for p := \frac{n+4}{n+2}, q := \frac{n}{n+2} one has 0 < q < p < 2,

p+q = 2, and that f := (\frac{ρ}{ρ_{0}} )^{n+2} is strictly positive on S^{n−1}, with ∫ f dμ_{n−1} = 1. Therefore the conclusion is a consequence of the Cauchy-Schwarz inequality,

1 = ∫ f^{\frac{p}{2} + \frac{q}{2}} dμ_{n−1} ⩽ (∫ f^{p} dμ_{n−1})^{\frac{1}{2}} (∫ f^{q} dμ_{n−1})^{\frac{1}{2}} ,

where equality holds if and only if f^{\frac{p}{2}} = cf^{\frac{q}{2}} for some constant c. Because of p ≠ q, the latter is equivalent to f = 1.

Remark 5.2. For any compact set K in ℝ^{n}, one has

(∫_{K} |x|^{2} dx)^{2} ⩽ ∫_{K} |x|^{4} dx ∫_{K} dx,

by the Cauchy-Schwarz inequality. Theorem 5.1 tells that, for compact sets K ⊆ ℝ^{n} which are star shaped with respect to the origin, this inequality can be improved to

(∫_{K} |x|^{2} dx)^{2} ⩽ c_{0} ∫_{K} |x|^{4} dx ∫_{K} dx,

with

c_{0} = \frac{n^{2}}{\frac{σ_{n−1}}{n+4} ρ_{0}^{n+4} \frac{σ_{n−1}}{n} ρ_{0}^{n}} = \frac{n(n + 4)}{(n + 2)^{2}} < 1.

This conclusion makes it clear that the inequality of Theorem 5.1 is not true for arbitrary compact subsets of ℝ^{n}. (Take, e.g., thin spherical shells!)

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