Moment inequalities and central limit properties of isotropic convex bodies

Ulrich Brehm, Peter Hinow, Hendrik Vogt, Jürgen Voigt^{⋆}

Fachrichtung Mathematik, Technische Universität Dresden, D-01062 Dresden, Germany

e-mail: brehm@math.tu-dresden.de, vogt@math.tu-dresden.de,
voigt@math.tu-dresden.de

Abstract The object of our investigations are isotropic convex bodies K ⊆ ℝ^{n}, centred at the origin and normed to volume one, in arbitrary dimensions. We show that a certain subset of these bodies—specified by bounds on the second and fourth moments—is invariant under forming ‘expanded joins’.

Considering a body K as above as a probability space and taking u ∈ S^{n−1}, we define random variables X_{K,u} = x · u on K. It is known that for subclasses of isotropic convex bodies satisfying a ‘concentration of mass property’, the distributions of these random variables are close to Gaussian distributions, for high dimensions n and ‘most’ directions u ∈ S^{n−1}. We show that this ‘central limit property’, which is known to hold with respect to convergence in law, is also true with respect to L_{1}-convergence and L_{∞}-convergence of the corresponding densities.

Mathematics Subject Classification (2000): 52A20, 60F25, 26B25 Key words: convex body, isotropic, moment inequality, central limit property, logarithmically concave function, concentration of mass

1 Introduction

Let K ⊆ ℝ^{n} be a convex body, i.e. a convex compact set with nonempty interior. We say that K is isotropic if its ellipsoid of inertia is a Euclidean ball, i.e.,

L_{K}^{2} := ∫_{K}(x · u)^{2} dx

is independent of the unit vector u ∈ S^{n−1}. The number L_{K} is called the radius of inertia of the body K. We say that K is normed if its centroid is at the origin and its n-dimensional volume λ_{n}(K) equals 1. Note that every convex body with nonempty interior has an affine image that is normed and isotropic.

⋆ partly supported by the DFG

For more information on isotropic convex bodies, related notions, and further properties and applications we refer to [2], [7], [8].

The set of all normed isotropic convex bodies in ℝ^{n} is denoted by K_{n}, and we define

K := ⋃_{n=1}^{∞} K_{n}.

We regard a body K ∈ K_{n}, with the measure λ_{n} restricted to K, as a probability space. For each 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 the distribution of this random variable is given by

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

In [4] it was shown that for Euclidean balls and for cubes these densities are close to Gaussian densities for large dimensions and most directions u, in the sense made precise below. Further, it was shown in [1] that an analogous property holds for the distribution functions of the random variables X_{K,u} for classes of symmetric convex bodies satisfying the ‘concentration of mass property’ described in Theorem 4.1(i). Also, in [10] it was observed that this concentration of mass property is a consequence of inequality (1.3) below, and the application of [11] was shown to yield a ‘central limit property’.

In order to make this more precise, we recall the following definition from [4]. Let M ⊆ K be a set of normed isotropic convex bodies. We say that M 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.

(1.1)

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 fact, the central limit property was expressed in [4] slightly differently—but equivalently—by requiring

sup_{K∈K_{n}∩M} E ∥φ_{K,u} − g_{L_{K}^{2}} ∥_{1} → 0 as n → ∞,

where E denotes expectation with respect to the probability measure μ_{n−1}.)

On the one hand, for the examples in [4] the central limit property (1.1), which involves the L_{1}-distance of densities, was shown. On the other hand, in the more general results [11], [1] only weaker forms of closeness were obtained. This observation was one of the starting points of this paper. As our main result we show that the central limit property (1.1) can be derived from a seemingly weaker form of this property. This is mainly due to the logarithmic concavity of the densities φ_{K,u}.

We use the following notation. For K ∈ K_{n} let

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

denote the second and fourth moment of the body K, respectively. Let T_{n} ⊆ K_{n} be the set of all normed isotropic bodies K that satisfy

m_{2}(K) ⩽ m_{2}(Δ_{n}) ,  (1.2)
\frac{m_{4}(K)}{m_{2}(K)^{2}} ⩽ \frac{m_{4}(Δ_{n})}{m_{2}(Δ_{n})^{2}} ,  (1.3)

where Δ_{n} denotes the normed regular n-simplex in ℝ^{n}. Let

T := ⋃_{n=1}^{∞} T_{n}.  (1.4)

Inequalities (1.2), (1.3)appear in [10] and [3], with the right hand sides written as explicit expressions depending on n. The set T is invariant under certain operations such as forming cones, cartesian products, joins (cf. [10]), and p-products (cf. [3]). As a consequence, T contains all ℓ_{p}^{n}-balls (normed to volume 1). Particular p-products are the join (p = 1) and the cartesian product (p = ∞).

It has been shown in [9] that, for n = 2, inequality (1.2) holds for all K ∈ K_{2}. It appears that one does not know a body in K not satisfying inequalities (1.2) and (1.3).

In Section 2 we introduce the expanded join of two bodies in K and show that T is invariant under forming normed isotropic expanded joins. Sections 3 and 4 are independent of Section 2.

Section 3 is devoted to showing properties of logarithmically concave functions. The main result, Theorem 3.3, states that closeness of logarithmically concave densities to Gaussian densities can be expressed in several equivalent ways: in L_{∞}-norm or L_{1}-norm of the difference of the densities, or in L_{∞}-norm of the difference of the corresponding distribution functions.

In Section 4 we show that a set M ⊆ K satisfying a concentration of mass property (see property (i) of Theorem 4.1) will also satisfy the central limit property defined in (1.1). Our main contribution is showing that in the case treated here a result of von Weizsäcker [11] can be reinforced using the results of Section 3. The general result for M is then applied to the class T defined above.

2 The expanded join

Continuing the investigations in [10], [3], we introduce yet another operation on convex bodies, the expanded join.

Let n_{0}, n_{1} ∈ ℕ_{0}, K_{j} ⊆ ℝ^{n_{j}} (j = 0, 1), and n = n_{0} + n_{1} + 1. We define the (unnormed) expanded join of K_{0} and K_{1} by

K_{0}ˆ•K_{1} = conv ((K_{0} × {0} × {0}) ∪ ({0} × K_{1} × {1}))

= {((1 − λ)x′, λx′′, λ); x′ ∈ K_{0}, x′′ ∈ K_{1}, λ ∈ [0, 1]}

= ⋃_{λ∈[0,1]} K̃_{λ} ⊆ ℝ^{n},

where K̃_{λ} = (1 − λ)K_{0} × (λK_{1}) × {λ}.

The set of extreme points is

ext(K_{0}ˆ•K_{1}) = (ext(K_{0}) × {0} × {0}) ∪ ({0} × ext(K_{1}) × {1}),

thus in particular the expanded join of two simplices is a simplex (merely count the vertices). In the special case n_{1} = 0, K_{1} = Δ_{0} = ℝ^{0}, the expanded join K_{0}ˆ•K_{1} is a cone with base K_{0} × {0}. Finally note that the orthogonal projection P_{n}: ℝ^{n} → ℝ^{n−1} (omit the last component) maps K_{0}ˆ•K_{1} onto the join of K_{0} and K_{1} as treated in [10]. This motivates the notion ‘expanded join’.

In the following we only consider n_{0}, n_{1} ⩾ 1. Our first aim is to find an affine image of K_{0}ˆ•K_{1} that is normed and isotropic. For the volume of K_{0}ˆ•K_{1} we have

λ_{n}(K_{0}ˆ•K_{1}) = ∫_{0}^{1} λ_{n−1}(K̃_{λ}) dλ = ∫_{0}^{1}(1 − λ)^{n_{0}}λ^{n_{1}} dλ

= B(n_{0} + 1, n_{1} + 1) = \frac{Γ (n_{0} + 1)Γ (n_{1} + 1)}{Γ (n_{0} + n_{1} + 2)} = \frac{n_{0}!n_{1}!}{n!} .

The centroid of K_{0}ˆ•K_{1} is at r := (0, …, 0, λ_{c}) where

λ_{c} = \frac{1}{λ_{n}(K_{0}ˆ•K_{1})} ∫_{0}^{1} ∫K̃_{λ} λ d(x′, x′′) dλ = \frac{n!}{n_{0}!n_{1}!} ∫_{0}^{1}(1 − λ)^{n_{0}}λ^{n_{1}+1} dλ

= \frac{n!}{n_{0}!n_{1}!} \frac{n_{0}!(n_{1} + 1)!}{(n + 1)!} = \frac{n_{1} + 1}{n + 1} .

Since K_{0} and K_{1} are isotropic, the function

u ↦ ∫_{K_{0}ˆ•K_{1}−r}(x · u)^{2} dx

is constant on S^{n_{0}−1} × {0} × {0} and {0} × S^{n_{1}−1} × {0}. Thus we obtain an isotropic image of K_{0}ˆ•K_{1} − r by applying a linear mapping T of the form

T = diag(\underbrace{α_{0}, …, α_{0}}_{n_{0}−times}, \underbrace{α_{1}, …, α_{1}}_{n_{1}−times}, α_{2}).

This means that we have to determine α_{0}, α_{1}, α_{2} > 0 such that

\begin{aligned}\frac{1}{n_{0}} ∫_{T (K_{0}ˆ•K_{1}−r)} |x′|^{2} d(x′, x′′, λ) & = \frac{1}{n_{1}} ∫_{T (K_{0}ˆ•K_{1}−r)} |x′′|^{2} d(x′, x′′, λ) \\ & = ∫_{T (K_{0}ˆ•K_{1}−r)} λ^{2} d(x′, x′′, λ). \end{aligned} (2.1)

As the second moment in the x′-direction we obtain

\frac{1}{n_{0}} ∫_{K_{0}ˆ•K_{1}−r} |x′|^{2} d(x′, x′′, λ) = \frac{1}{n_{0}} ∫_{0}^{1} ∫K̃_{λ} |x′|^{2} d(x′, x′′) dλ
= \frac{1}{n_{0}} ∫_{0}^{1} ∫_{K_{0}} |x′|^{2} dx′(1 − λ)^{n_{0}+2}λ^{n_{1}} dλ = a_{0}m_{2}(K_{0}),

where a_{0} = \frac{(n_{0}+2)!n_{1}!}{n_{0}(n+2)!}. Analogously we have

\frac{1}{n_{1}} ∫_{K_{0}ˆ•K_{1}−r} |x′′|^{2} d(x′, x′′, λ) = a_{1}m_{2}(K_{1})

and

∫_{K_{0}ˆ•K_{1}−r} λ^{2} d(x′, x′′, λ) = ∫_{0}^{1} ∫K̃_{λ} d(x′, x′′)(λ − λ_{c})^{2} dλ = a_{2},

where a_{1}, a_{2} > 0 depend only on n_{0}, n_{1}. From (2.1) we now obtain the condition

α_{0}^{n_{0}+2}α_{1}^{n_{1}}α_{2}a_{0}m_{2}(K_{0}) = α_{0}^{n_{0}}α_{1}^{n_{1}+2}α_{2}a_{1}m_{2}(K_{1}) = α_{0}^{n_{0}}α_{1}^{n_{1}}α_{2}^{3}a_{2}

for α_{0}, α_{1}, and α_{2}, which simplifies to

α_{0}^{2}a_{0}m_{2}(K_{0}) = α_{1}^{2}a_{1}m_{2}(K_{1}) = α_{2}^{2}a_{2}.  (2.2)

From the requirement λ_{n}(T (K_{0}ˆ•K_{1} − r)) = 1 we obtain the condition

α_{0}^{n_{0}}α_{1}^{n_{1}}α_{2} = \frac{n!}{n_{0}!n_{1}!} .  (2.3)

Solving (2.2) for α_{0} and α_{1} and inserting this into (2.3) yields

α_{2} = (\frac{n!}{n_{0}!n_{1}!} (\frac{a_{0}}{a_{2}} m_{2}(K_{0}))^{\frac{n_{0}}{2}} (\frac{a_{1}}{a_{2}} m_{2}(K_{1}))^{\frac{n_{1}}{2}} )^{\frac{1}{n}} .  (2.4)

For our further considerations we need not compute α_{0}, α_{1} and α_{2} more explicitly. We define the normed isotropic expanded join by

K_{0} • K_{1}
= conv ((α_{0}K_{0} × {0} × {−α_{2}λ_{c}}) ∪ ({0} × α_{1}K_{1} × {α_{2}(1 − λ_{c})}))
= {(α_{0}(1 − λ)x′, α_{1}λx′′, α_{2}(λ − λ_{c})); x′ ∈ K_{0}, x′′ ∈ K_{1}, λ ∈ [0, 1]}.

Theorem 2.1. The set T defined in (1.4) is invariant under forming normed isotropic expanded joins.

Proof. We have to show the persistence of inequalities (1.2) and (1.3). We may assume n_{0}, n_{1} ⩾ 1 since the case of cones has been treated in [10; Sect. 2]. For the second moment of K_{0} • K_{1} we obtain, using (2.1), (2.3), and (2.4),

m_{2}(K_{0} • K_{1}) = n ∫_{K_{0}•K_{1}} λ^{2} d(x′, x′′, λ) = α_{0}^{n_{0}}α_{1}^{n_{1}}α_{2}^{3} na_{2}

= \frac{n!}{n_{0}!n_{1}!} (\frac{n!}{n_{0}!n_{1}!} (\frac{a_{0}}{a_{2}} m_{2}(K_{0}))^{\frac{n_{0}}{2}} (\frac{a_{1}}{a_{2}} m_{2}(K_{1}))^{\frac{n_{1}}{2}} )^{\frac{2}{n}} na_{2}.

Thus the validity of (1.2) for K_{0} and K_{1} implies

m_{2}(K_{0} • K_{1}) ⩽ m_{2}(Δ_{n_{0}} • Δ_{n_{1}}) = m_{2}(Δ_{n}),

which is (1.2) for K_{0} • K_{1}.

The fourth moment is computed as

∫_{K_{0}•K_{1}} |x|^{4} dx = ∫_{K_{0}•K_{1}}(|x′|^{2} + |x′′|^{2} + λ^{2})^{2} dx
= α_{0}^{n_{0}+4}α_{1}^{n_{1}}α_{2}b_{0} m_{4}(K_{0}) + α_{0}^{n_{0}}α_{1}^{n_{1}+4}α_{2}b_{1} m_{4}(K_{1})
+ α_{0}^{n_{0}}α_{1}^{n_{1}}α_{2}^{5}b_{2} + α_{0}^{n_{0}+2}α_{1}^{n_{1}+2}α_{2}b_{01}m_{2}(K_{0})m_{2}(K_{1})

+ α_{0}^{n_{0}+2}α_{1}^{n_{1}}α_{2}^{3}b_{02}m_{2}(K_{0}) + α_{0}^{n_{0}}α_{1}^{n_{1}+2}α_{2}^{3}b_{12}m_{2}(K_{1}),

where again b_{0}, b_{1}, b_{2}, b_{01}, b_{02}, b_{12} > 0 depend only on n_{0}, n_{1}. Using (2.3)as well as

\frac{1}{n}m_{2}(K_{0} • K_{1}) = α_{0}^{n_{0}+2}α_{1}^{n_{1}}α_{2}a_{0} m_{2}(K_{0})
= α_{0}^{n_{0}}α_{1}^{n_{1}+2}α_{2}a_{1} m_{2}(K_{1})
= α_{0}^{n_{0}}α_{1}^{n_{1}}α_{2}^{3}a_{2}

appropriately, we obtain

\frac{m_{4}(K_{0} • K_{1})}{m_{2}(K_{0} • K_{1})^{2}} = c_{0} \frac{m_{4}(K_{0})}{m_{2}(K_{0})^{2}} + c_{1} \frac{m_{4}(K_{1})}{m_{2}(K_{1})^{2}} + c_{2},

with suitable c_{0}, c_{1}, c_{2} > 0 depending only on n_{0}, n_{1}. Similarly as above for the second moment, the validity of (1.3) for K_{1}, K_{2} implies (1.3) for K_{0} • K_{1}. ⊓ ⊔ 3 Inequalities for logarithmically concave functions

The assertion of Lemma 3.2(b) below will be needed in Section 4. Here we prove rather general explicit inequalities for logarithmically concave functions, which should be of independent interest. In Section 4 we would only need a qualitative estimate for a specific case.

We recall that a function f: I → [0, ∞) is called logarithmically concave if ln f is concave on an interval I ⊆ ℝ (with the usual convention ln 0 = −∞). For a ∈ ℝ we denote a^{+} := max{a, 0}.

Lemma 3.1. Let 0 ⩽ f ∈ L_{1}([0, ∞)) be logarithmically concave, f(0) > 0, ∥f∥_{1} > 0, a := \frac{∥f∥_{1}}{f(0)} . Let g: [0, ∞) → ℝ be increasing. Assume that

t_{1} := sup{t ⩾ 0; f ⩾ g on [0, t]} < ∞.

Then

∫_{0}^{t_{1}}(f − g)(t) dt ⩾ ∫_{0}^{∞} (f(0)e^{−t/a} − g(t))^{+}dt.

Proof. Let f̃(t) := f(0)e^{−t/a} (t ⩾ 0) and

t_{2} := sup{t ⩾ 0; f̃ ⩾ g on [0, t]} (∈ [0, ∞]).

Then f̃ ⩾ g on [0, t_{2}), and f̃ ⩽ g on (t_{2}, ∞) since f̃− g is decreasing. We have to show ∫_{0}^{t_{2}}(f̃− g)(t) dt ⩽ ∫_{0}^{t_{1}}(f − g)(t) dt.

In case f = f̃ there is nothing to show, so we assume f ≠ f̃. Note that ∫ f̃ = ∫ f by the definition of a. Thus, f − f̃ changes sign at least once. Since ln f − ln f̃ is concave and f(0) = f̃(0), there exists t_{0} > 0 such that f ⩾ f̃ on [0, t_{0}) and f ⩽ f̃ on (t_{0}, ∞). This implies ∫_{0}^{s} f̃(t) dt ⩽ ∫_{0}^{s} f(t) dt for all s ⩾ 0 and hence

∫_{0}^{t_{2}}(f̃− g)(t) dt ⩽ ∫_{0}^{t_{2}}(f − g)(t) dt.

Assume first that t_{2} ⩽ t_{0}. Then t_{2} ⩽ t_{1} ⩽ t_{0}, so we obtain the assertion since f − g ⩾ 0 on (t_{2}, t_{1}). Now assume t_{2} > t_{0}. Then t_{0} ⩽ t_{1} ⩽ t_{2}. We have f(⩽ f̃) ⩽ f(0) on (t_{0}, ∞), so f and thus f − g is decreasing on (t_{0}, ∞). By the definition of t_{1}, this implies f −g ⩽ 0 on (t_{1}, t_{2}), and we obtain the assertion. ⊓ ⊔

Lemma 3.2. Let 0 ⩽ f ∈ L_{1}(ℝ) be logarithmically concave, ∥f∥_{1} ⩾ 1, g ∈ L_{1}(ℝ) ∩ L_{∞}(ℝ), and F, G antiderivatives of f, g, respectively.

(a) Let α > 1. If ∥F − G∥_{∞} ⩽ \frac{1}{2} − \frac{1+ln α}{2α} then ∥f∥_{∞} ⩽ α ess sup g.

(b) If g is Lipschitz continuous with Lipschitz constant L then

∥f − g∥_{∞} ⩽ 2\sqrt{L + ∥f∥_{∞}^{2}}∥F − G∥∞^{\frac{1}{2}}.

(c) Let α > 1. If ∥F − G∥_{∞} ⩽ \frac{1}{2} − \frac{1+ln α}{2α}, ∥g∥_{1} ⩽ 1, and g is Lipschitz continuous with Lipschitz constant L then

∥f − g∥_{∞} ⩽ 2\sqrt{L}\sqrt{1 + α^{2}}∥F − G∥∞^{\frac{1}{2}}.

Proof. We will make use of the inequality | ∫_{a}^{b}(f(t) − g(t)) dt| ⩽ 2∥F − G∥_{∞}, which is valid for all a, b ∈ ℝ.

(a) Observe that M := ess sup g ⩾ 0. Without restriction assume that f(0) = ∥f∥_{∞} > M. Let I be the maximal open interval containing 0 with f ⩾ g on I. Then

2∥F − G∥_{∞} ⩾ ∫_{I}(f − g)(t) dt ⩾ ∫_{−∞}^{∞}(f(t) − M)^{+}dt.

By Lemma 3.1, with g = M, we have

∫_{0}^{∞}(f(t) − M)^{+}dt ⩾ ∫_{0}^{∞}(f(0)e^{−t/a} − M)^{+}dt,

where a := ∫_{0}^{∞} f(t) dt/f(0). The above inequality makes sense in the case a = 0, too; then both sides are 0. In the same way we obtain

∫_{−∞}^{0}(f(t) − M)^{+}dt ⩾ ∫_{−∞}^{0}(f(0)e^{t/b} − M)^{+}dt,

with b := ∫_{−∞}^{0} f(t) dt/f(0).

Let t_{0} := a ln \frac{f(0)}{M} and s_{0} := −b ln \frac{f(0)}{M}. Then f(0)e^{−t_{0}/a} = f(0)e^{s_{0}/b} = M. Therefore,

∫_{−∞}^{∞}(f(t) − M)^{+}dt ⩾ ∫_{s_{0}}^{0}(f(0)e^{t/b} − M) dt + ∫_{0}^{t_{0}}(f(0)e^{−t/a} − M) dt
= b(f(0) − M) − a(M − f(0)) − (t_{0} − s_{0})M
= (a + b)(f(0) − M − M ln \frac{f(0)}{M}).

Since a + b = ∥f∥_{1}/f(0) ⩾ \frac{1}{f(0)} we arrive at

2∥F − G∥_{∞} ⩾ ∫_{−∞}^{∞}(f(t) − M)^{+}dt ⩾ 1 − \frac{M}{f(0)} (1 + ln \frac{f(0)}{M}) .

Using 2∥F − G∥_{∞} ⩽ 1 − \frac{1}{α}(1 + ln α), we deduce that ∥f∥_{∞} = f(0) ⩽ αM.

(b) We have to estimate |f(x) − g(x)| for all x ∈ ℝ. Since the assumptions are translation invariant it suffices to consider x = 0. First assume f(0) ⩽ g(0). We suppose that f is decreasing on [0, ∞). (If this is not the case then f is increasing on (−∞, 0]; the latter case is treated analogously.) Let η := \frac{g(0)−f(0)}{L}. By the Lipschitz continuity of g we have

g(t) − f(t) ⩾ g(0) − Lt − f(0) = L(η − t) (t ⩾ 0),

and thus

2∥F − G∥_{∞} ⩾ ∫_{0}^{η}(g(t) − f(t)) dt ⩾ ∫_{0}^{η} L(η − t) dt = \frac{(g(0) − f(0))^{2}}{2L} ,

Moment inequalities, central limit properties, isotropic convex bodies  9

PSfrag replacements h f̃ g̃

Figure 1

or

g(0) − f(0) ⩽ 2\sqrt{L∥F − G∥_{∞}}.

Now assume h := f(0) − g(0) > 0. We have to show ∥F − G∥_{∞} ⩾ \frac{h^{2}}{4(L+∥f∥_{∞}^{2})}.

Let g̃(t) := g(0) + L|t| (t ∈ ℝ). Let I (Ĩ) be the maximal interval containing 0 with f ⩾ g on I (f ⩾ g̃ on Ĩ). Then, by the Lipschitz continuity of g,

2∥F − G∥_{∞} ⩾ ∫_{I}(f − g)(t) dt ⩾ ∫Ĩ(f − g̃)(t) dt.

Let a, b as in the proof of (a), and define f̃ by f̃(t) := f(0)e^{−t/a} (t ⩾ 0) and f̃(t) := f(0)e^{t/b} (t ⩽ 0). The first inequality in the following estimate follows from Lemma 3.1, whereas the second inequality is illustrated by Figure 1 (the area between the graphs of g̃ and f̃ contains the shaded area):

∫Ĩ(f − g̃)(t) dt ⩾ ∫_{−∞}^{∞} (f̃(t) − g̃(t))^{+}dt

⩾ ∫_{0}^{∞} (h − \frac{f(0)}{a}t − Lt)^{+}dt + ∫_{−∞}^{0} (h + \frac{f(0)}{b}t + Lt)^{+}dt

= \frac{h^{2}}{2(L + \frac{f(0)}{a})} + \frac{h^{2}}{2(L + \frac{f(0)}{b})}.

Recall that c := a + b = ∥f∥_{1}/f(0). Thus,

2∥F − G∥_{∞} ⩾ min_{0⩽a⩽c} \frac{h^{2}}{2} (\frac{1}{L + \frac{f(0)}{a}} + \frac{1}{L + \frac{f(0)}{c−a}} ) .

The mininum is attained at a = 0 and at a = c, so

2∥F − G∥_{∞} ⩾ \frac{h^{2}}{2(L + \frac{f(0)^{2}}{∥f∥_{1}} )} ⩾ \frac{h^{2}}{2(L + ∥f∥_{∞}^{2})},

which was to be shown.

(c) is a direct consequence of (a) and (b) since ∥g∥_{∞} ⩽ \sqrt{L}. ⊓ ⊔

We will apply Lemma 3.2 in form of the next theorem, where G_{σ^{2}} denotes the Gaussian distribution function with variance σ^{2},

G_{σ^{2}}(t) = \frac{1}{\sqrt{2πσ^{2}}} ∫_{−∞}^{t} exp (−\frac{s^{2}}{2σ^{2}} ) ds.

Theorem 3.3. There exists an increasing function β_{1}: (0, ∞) → (0, 2] with

β_{1}(t) = O(t(− ln t)^{\frac{1}{2}} ) (0 < t ⩽ \frac{1}{\sqrt{2π}} )

with the following property. Let 0 ⩽ f ∈ L_{1}(ℝ) be logarithmically concave, ∥f∥_{1} = 1, and F the distribution function of f. Let σ > 0 and assume that ∥f∥_{∞} ⩽ \frac{1}{σ}. Then

∥F − G_{σ^{2}}∥_{∞} ⩽ ∥f − g_{σ^{2}}∥_{1} ⩽ β_{1}(σ∥f − g_{σ^{2}}∥_{∞}) ⩽ β_{1}(\sqrt{5}∥F − G_{σ^{2}}∥∞^{\frac{1}{2}}),

∥f − g_{σ^{2}}∥_{∞} ⩽ \frac{\sqrt{5}}{σ}∥F − G_{σ^{2}}∥∞^{\frac{1}{2}}.

Proof. The first inequality is clear; the second one is proved in [4; Prop. 2.5]. The third inequality is a consequence of the last one. Note that g_{σ^{2}} is Lipschitz continuous with constant \frac{1}{\sqrt{2πe}σ^{2}} < \frac{1}{4σ^{2}} . Thus, the last inequality is a direct consequence of Lemma 3.2(b). ⊓ ⊔

Remark 3.4.If f is logarithmically concave then the assumption ∥f∥_{∞} ⩽ \frac{1}{σ} in the above theorem is in particular satisfied if ∫_{−∞}^{∞} tf(t) dt = 0 and σ^{2} is the variance of f. This follows from [6; Thm. 8, eqn. (7)], applied with φ(t) = t^{2}.

4 A central limit theorem

The main result of this section (and the whole paper) is the following. As in Section 3, G_{σ^{2}} denotes the Gaussian distribution function with variance σ^{2}. Also,

Φ_{K,u}(t) = λ_{n}({x ∈ K; x · u ⩽ t})

is the distribution function of the density φ_{K,u}, for K ∈ K_{n}, u ∈ S^{n−1}.

Theorem 4.1. Let M ⊆ K be a set of normed isotropic convex bodies. Consider the following properties:

(i) sup_{K∈K_{n}∩M} λ_{n}({x ∈ K; ||x|^{2} − nL_{K}^{2} | > εnL_{K}^{2}}) → 0 as n → ∞, for all ε > 0;

(ii) sup_{K∈K_{n}∩M} μ_{n−1}({u ∈ S^{n−1}; ∥Φ_{K,u} − G_{L_{K}^{2}} ∥_{∞} > ε}) → 0 as n → ∞,

for all ε > 0; (iii) 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, i.e., M has the central limit property (1.1); (iv) sup_{K∈K_{n}∩M} μ_{n−1}({u ∈ S^{n−1}; ∥φ_{K,u} − g_{L_{K}^{2}} ∥_{∞} > ε}) → 0 as n → ∞,

for all ε > 0. Then (i) ⇒ (ii) ⇔ (iii) ⇒ (iv). If sup_{K∈M} L_{K} < ∞ then also (iv) ⇒ (iii).

Theorem 4.2. The set T defined in (1.4) satisfies properties (i) – (iv) of Theorem 4.1.

Remark 4.3.In [1; Thm. 4] the implication (i) ⇒ (ii) of Theorem 4.1 is shown for classes of centrally symmetric bodies, with quantitative estimates. By use of our Theorem 3.3, this result can be reinforced to a quantitative result concerning L_{1}-norms and L_{∞}-norms of the difference of the corresponding densities.

Before proceeding to the proof of Theorem 4.1 we want to recall a result of von Weizsäcker [11] which will be an essential ingredient of our proof.

For K ∈ K_{n} we define probability measures P_{K} and P̃_{K} on the σ-algebra B(ℝ^{n}) of Borel sets of ℝ^{n} by

P_{K}(B) := λ_{n}(B ∩ K),
P̃_{K}(B) := P_{K}(L_{K}B) (B ∈ B(ℝ^{n})).

For the marginal densities and the corresponding distribution functions of the measure P̃_{K} we obtain

\begin{aligned}φ̃_{K,u}(t) & = L_{K}φ_{K,u}(L_{K}t), \\ Φ̃_{K,u}(t) & = Φ_{K,u}(L_{K}t) (t ∈ ℝ, u ∈ S^{n−1}), \end{aligned}  (4.1)

respectively.

Since K is isotropic the covariance matrix C of P̃_{K} is computed as

C = (∫_{ℝ^{n}} x_{i}x_{j} dP̃_{K}(x))_{i,j=1,...,n} = L_{K}^{−2} (∫_{K} x_{i}x_{j} dx)_{i,j=1,...,n} = E_{n}.

Hence the conditions [11; (9) and (10)], i.e.

trace C = O(n),
∑_{i,j=1}^{n} C_{ij}^{2} = o(n^{2}),

are satisfied. Further, property (i) in Theorem 4.1 can be expressed as

sup_{K∈K_{n}∩M} P̃_{K}(\{x ∈ ℝ^{N}; |\frac{|x|}{\sqrt{n}} − 1| ⩾ ε\}) → 0 as n → ∞, for all ε > 0,

i.e., the condition [11; (11)] is satisfied.

In order to describe the topology of convergence in law on Prob(ℝ), the probability measures on B(ℝ), we use the Lévy metric; cf. [5; Section 8.1, Theorem 3]. For P, Q ∈ Prob(ℝ), with distribution functions F, G, the Lévy distance L(P, Q) is defined by

L(P, Q) := inf {ε > 0; F (t − ε) − ε ⩽ G(t) ⩽ F (t + ε) + ε for all t ∈ ℝ}.

Expressed with this metric, von Weizsäcker’s result implies that, in case property (i) in Theorem 4.1 holds, we have

sup_{K∈K_{n}∩T} μ_{n−1} ({u ∈ S^{n−1}; L(φ̃_{K,u}λ_{1}, g_{1}λ_{1}) > ε}) → 0  (4.2)

as n → ∞, for all ε > 0. (This follows from [11; Corollary 1]. Also, one has to use that, due to the concentration of mass, the mixed Gaussian can be replaced by the Gaussian with variance 1; cf. [11; p. 316, line 10].)

The above is the main part of the proof of the implication (i)⇒(ii) in Theorem 4.1. We further need the following preliminary facts.

Lemma 4.4. Let c > 0, and let P, Q ∈ Prob(ℝ), with associated distribution functions F, G, respectively. Assume that F, G are Lipschitz continuous with Lipschitz constant c. Then

∥F − G∥_{∞} ⩽ (1 + c)L(P, Q).

Proof. It suffices to show G(t) − F (t) ⩽ (1 + c)L(P, Q) for all t ∈ ℝ. Let t ∈ ℝ, ε > L(P, Q). Then

G(t) ⩽ F (t + ε) + ε ⩽ F (t) + cε + ε.

Letting ε → L(P, Q) we obtain G(t) − F (t) ⩽ (1 + c)L(P, Q). ⊓ ⊔

Remark 4.5.For all K ∈ K we have

L_{K} ⩾ \frac{1}{\sqrt{2πe}} .

Indeed, it is well-known that L_{K} ⩾ L_{B_{n}} for all K ∈ K_{n}, where B_{n} is the normed Euclidean ball (we have

nL_{K}^{2} = ∫_{K} |x|^{2}dx ⩾ ∫_{B_{n}} |x|^{2}dx = nL_{B_{n}}^{2},

since the norm of a vector in K \ B_{n} is always greater than the norm of a vector in B_{n} \ K). The radius of inertia of the normalised ball is given by (see [4; Section 3], easy computation)

L_{B_{n}}^{2} = \frac{Γ (\frac{n}{2} + 1)^{\frac{2}{n}}}{(n + 2)π} ⩾ \frac{n\sqrt{2π \frac{n}{2}} ^{\frac{2}{n}}}{2e(n + 2)π} ⩾ \frac{1}{2πe} .

Proof of Theorem 4.1. We make use of the fact that the densities φ_{K,u} are logarithmically concave. Indeed, by Brunn’s theorem (cf. [2; Theorem 5.1]) the functions φ_{K,u}(·)^{\frac{1}{n−1}} are concave on their support. Since the logarithm is increasing and concave, ln φ_{K,u} = (n − 1) ln(φ_{K,u}^{\frac{1}{n−1}} ) is concave on ℝ.

(i) ⇒ (ii). From (4.1) and Remark 3.4 we know that ∥φ̃_{K,u}∥_{∞} ⩽ 1 for all K ∈ K. By (4.1), Lemma 4.4, and (4.2) we obtain

sup_{K∈K_{n}∩M} μ_{n−1}({u ∈ S^{n−1}; ∥Φ_{K,u} − G_{L_{K}^{2}} ∥_{∞} > ε})

= sup_{K∈K_{n}∩M} μ_{n−1}({u ∈ S^{n−1}; ∥Φ̃_{K,u} − G_{1}∥_{∞} > ε})

⩽ sup_{K∈K_{n}∩M} μ_{n−1}(\{u ∈ S^{n−1}; L(φ_{K,u}λ_{1}, g_{L_{K}^{2}} λ_{1}) > \frac{ε}{2}\}) → 0

as n → ∞.

(ii) ⇔ (iii) is a direct consequence of the first chain of inequalities in Theorem 3.3, and Remark 3.4.

(ii) ⇒ (iv). Because of Remark 3.4, we can apply the last inequality of Theorem 3.3 to f = φ_{K,u}, with σ = L_{K} ⩾ \frac{1}{\sqrt{2πe}} by Remark 4.5. Thus we have

sup_{K∈K_{n}∩M} μ_{n−1}({u ∈ S^{n−1}; ∥φ_{K,u} − g_{L_{K}^{2}} ∥_{∞} > ε})

⩽ sup_{K∈K_{n}∩M} μ_{n−1}(\{u ∈ S^{n−1}; \sqrt{5}\sqrt{2πe}\sqrt{∥Φ_{K,u} − G_{L_{K}^{2}} ∥_{∞}} > ε\})

= sup_{K∈K_{n}∩M} μ_{n−1}(\{u ∈ S^{n−1}; ∥Φ_{K,u} − G_{L_{K}^{2}} ∥_{∞} > \frac{ε^{2}}{10πe}\}) → 0

as n → ∞.

(iv) ⇒ (iii) under the additional assumption sup_{K∈M} L_{K} < ∞. This is similar to the proof of ‘(ii) ⇒ (iv)’ above. (Instead of the last inequality of Theorem 3.3, use the second one: ∥φ_{K,u} − g_{L_{K}^{2}} ∥_{1} ⩽ β_{1}(L_{K}∥φ_{K,u} − g_{L_{K}^{2}} ∥_{∞}).) ⊓ ⊔

Proof of Theorem 4.2. As noted in [10; eqn. (1.1)], inequality (1.3) implies property (i) of Theorem 4.1 for M = T . More precisely, for all ε > 0 we have

sup_{K∈T_{n}}λ_{n}({x ∈ K; ||x|^{2} − nL_{K}^{2} | ⩾ εnL_{K}^{2}})

⩽ \frac{1}{ε^{2}} (\frac{(n + 1)(n + 2)}{(n + 3)(n + 4)} (1 + \frac{2}{n} + \frac{6}{n + 1}) − 1) → 0

as n → ∞. Thus the statements are a consequence of Theorem 4.1. ⊓ ⊔

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