Moment estimates for convex measures

Let $p\geq 1$, $\eps>0$, $r\geq (1+\eps) p$, and $X$ be a $(-1/r)$-concave random vector in $\R^n$ with Euclidean norm $|X|$. We prove that $(\E |X|^{p})^{1/{p}}\leq c (C(\eps) \E|X|+\sigma_{p}(X))$, where $\sigma_{p}(X)=\sup_{|z|\leq 1}(\E||^{p})^{1/p}$, $C(\eps)$ depends only on $\eps$ and $c$ is a universal constant. Moreover, if in addition $X$ is centered then $(\E |X|^{-p})^{-1/{p}}\geq c(\eps) (\E|X| - C \sigma_{p}(X))$.


Introduction
Let X be a random vector with values in a finite dimensional Euclidean space E with Euclidean norm | • | and scalar product •, • .For any p > 0, we define the weak p-th moment of X by Clearly (E|X| p ) 1/p ≥ σ p (X) and by Hölder's inequality, (E|X| p ) 1/p ≥ E|X|.In this paper we are interested in reversed inequalities of the form for p ≥ 1 and constants C 1 and C 2 .This is known for some classes of distributions and the question has been studied in a more general setting (see [19] and references therein) and our objective in this paper is to describe new classes for which the relationship (1) is satisfied.
Let us recall some known results when (1) holds.It clearly holds for Gaussian vectors and it is not difficult to see that (1) is true for subgaussian vectors (see below for definitions) for every p ≥ 1, with C 1 and C 2 depending only on the subgaussian parameter.
Another example of such a class is the class of so-called log-concave vectors.A probability measure µ on R m is called log-concave if for all 0 < θ < 1 and for all compact subsets A, B ⊂ R m with positive measure one has µ((1 − θ)A + θB) ≥ µ(A) 1−θ µ(B) θ . (2) A random vector with a log-concave distribution is called log-concave.It is known that for every log-concave random vector X in a finite dimensional Euclidean space and any p > 0, where C > 0 is a universal constant.See Corollary 7 and references below.
In this paper we will consider the class of convex measures introduced by Borell.Let κ < 0. A probability measure µ on R m is called κ-concave if for all 0 < θ < 1 and for all compact subsets A, B ⊂ R m with positive measure one has µ(( A random vector with a κ-concave distribution is called κ-concave.Note that a log-concave vector is also κ-concave for any κ < 0. We show in Theorem 6 that for κ > −1, a κ-concave random vector satisfies (1) for all 0 < (1 + ε)p < −1/κ with C 1 and C 2 depending only on ε.
In fact, in Definition 1 we will introduce a general assumption on the distribution, called H(p, λ).The main result of the first part of the paper is Theorem 2 in which we show that this assumption is sufficient in order to have (1).In Theorem 5 we prove that convex measures satisfy this assumption.
One of the main applications of the relationship (1) consists in tail inequalities for P (|X| ≥ t E|X|).In Corollary 8 we show that for r > 2 and for a (−1/r)concave isotropic random vector X ∈ R n the above probability is bounded by c max{1,r/ √ n} t r/2 . From this bound we deduce that the empirical covariance matrix of a sample of size proportional to n is a good approximation of the covariance matrix of X, extending results of [1,2] from log-concave measures to convex measures.This provides thus a new class of random vectors satisfying such approximation.See Corollary 10 and the remark following it.
The second part of the paper deals with negative moments.We are looking for relationship of the form for p > 0 and constants C 1 and C 2 .We show in Theorem 14 that for κ > −1, an n-dimensional κ-concave random vector satisfies (4) for all 0 < (1 + ε)p < min{n/2, (−1/κ)} with C 1 and C 2 depending only on ε.As an application we show a small ball probability estimate for κ-concave random vectors.In the log-concave setting it was proved in [27].

Preliminaries
The space R m is equipped with the scalar product •, • , the Euclidean norm | • |, the unit ball B m 2 and the volume measure vol(•).The canonical basis is denoted by e 1 , e 2 , . . ., e m .A gauge or Minkowski functional • on R m is a non-negative function on R m satisfying: λx = λ x and x + y ≤ x + y for every x, y ∈ R m and every real λ ≥ 0 and such that x =0 if and only if x = 0.The dual gauge is defined for every x ∈ R m by x * = max{ x, t : t ≤ 1}.A body is a compact subset of R m with a non-empty interior.Any convex body K ⊂ R m containing the origin in its interior defines the gauge by x = inf{λ ≥ 0 : x ∈ λK}.It is called the Minkowski functional of K.If K ⊂ R m is a convex body containing the origin in its interior, the polar body For a linear subspace F ⊂ R n we denote the orthogonal projection on F by P F .Note that P F K For a random vector X in R n with a density g and a subspace F ⊂ R n , we denote the density of P F X by g F .
A random vector X in R m will be called non- Given a non-negative bounded function g on R m we introduce the following associated set.For any α ≥ 1, let where g ∞ = sup t∈R m |g(t)|.By g i , g i,j we denote independent standard Gaussian random variables, i.e. centered and of variance one.A standard Gaussian vector in R n is denoted by G, i.e.G = (g 1 , g 2 , ..., g n ).The standard Gaussian matrix is the matrix whose entries are i.i.d.standard Gaussian variables, i.e.Γ = {g i,j }.By γ p we denote the L p norm of g 1 .Note that γ p / √ p → 1/ √ e as p → ∞.We denote by µ n,k the Haar probability measure on the Grassmannian G n,k of k-dimensional subspaces of R n .
Recall that for a real number s, ⌈s⌉ denotes the smallest integer which is not less than s.
By C, C 0 , C 1 , C 2 , ..., c, c 0 c 1 , c 2 we denote absolute positive constants, whose values can change from line to line.
For two functions f and g we write f ∼ g if there are absolute positive constants c and C such that cf ≤ g ≤ Cf .

Convex probabilities
Let κ ≤ 1/m.A Borel probability measure µ on R m is called κ-concave if it satisfies (3).When κ = 0, this inequality should be read as (2) and it defines µ as a log-concave probability.
In this paper we will be interested in the case κ ≤ 0, which we consider from now on.
The class of κ-concave measures was introduced and studied by Borell.We refer to [9,10] for a general study and to [8] for more recent development.A κ-concave probability is supported on some convex subset of an affine subspace where it has a density.When the support generates the whole space, a characterization of Borell ([9,10]) states that the probability is absolutely continuous with respect to the Lebesgue measure and has a density g which is log-concave when κ = 0 and when κ < 0, is of the form where f : R m → (0, ∞] is a convex function.The class of m-dimensional κconcave probabilities is increasing as κ is decreasing.In particular a log-concave probability is κ-concave for any κ < 0. As we mentioned in the Introduction, a random vector with a κ-concave distribution is called κ-concave.Clearly, the linear image of a κ-concave probability is also κ-concave.Recall that any semi-norm of an m-dimensional vector with a κ-concave distribution has moments up to the order p < −1/κ (see [9] and Lemmas 21 and 22 below).Since we are interested in comparison of moments with the moment of order 1, we will always assume that −1 < κ ≤ 0.

Strong and weak moments
In this section we consider a random vector X in a finite dimensional Euclidean space E. Definition 1.Let p > 0, m = ⌈p⌉, and λ ≥ 1.We say that a random vector X in E satisfies the assumption H(p, λ) if for every linear mapping Remark.Let us give a first example of a random vector satisfying H(p, λ).
Let X be a random vector in an n-dimensional Euclidean space E, satisfying, for some ψ ≥ 1, Then X satisfies H(p, Cψ 2 ) for every p ≥ 1.For example, the standard Gaussian and Rademacher vectors satisfy the above condition with ψ being a numerical constant.More generally, if X is subgaussian, then X satisfies (7).
To prove that (7) implies H(p, Cψ 2 ), let p > 0, m = ⌈p⌉ and let A : E → R m be such that Y = AX is non-degenerate.We may assume that m ≥ 2. Clearly, because of the linear invariance of the property (7), we may also assume that Y = AX is isotropic.Thus (7) yields, where the last inequality follows from isotropicity of Y by applying (7) with p = 2, z i = A ⊤ e i , i ≤ m, and the Cauchy-Schwarz inequality.Now let us make the following general observation.Let p ≥ 1 and m = ⌈p⌉.Let Y be a random vector in an m-dimensional normed space with norm • .Since any m-dimensional norm can be estimated, up to a multiplicative constant, by the supremum over an exponential (in m) number of norm one linear forms, we deduce that where C ′ is a universal constant (see [20] Proposition 3.20).Combining this with (8) we conclude that The main result of this section states a relationship between weak and strong moments under the assumption H(p, λ).
Theorem 2. Let p > 0 and λ ≥ 1.If a random vector X in a finite dimensional Euclidean space satisfies H(p, λ), then where c is a universal constant.
The first step of the proof of Theorem 2 consists of showing that there exists some z such that (E( z, Y ) p + ) 1/p is small, with comparison to E|Y |.This is the purpose of the following lemma.
Lemma 3. Let Y be a random vector in R m .Let • 1 and • 2 be two gauges on R m and • * 1 and • * 2 be their dual gauges.Then for all p > 0, min Proof.Let r be the largest real number such that r t 1 ≤ t 2 for all t ∈ R m .By duality r is the largest number such that r w * The second step of the proof of Theorem 2 is contained in the next lemma.Lemma 4. Let n, m ≥ 1 be integers.Let p ≥ 1.Let X be an n-dimensional random vector and Γ be an n × m standard Gaussian matrix.Then where z p,+ = E ( z, X ) p + 1/p and C is a universal constant.
Proof.For every x, y ∈ R n , | x p,+ − y p,+ | ≤ |x − y|σ p (X).The classical Gaussian concentration inequality (see [12] or inequality (2.35) and Proposition 2.18 in [21]) gives that and implies (cf.[22], Statement 3.1) where C is a universal constant.Since G, X has the same distribution as |X| g 1 , we have Therefore The Gordon minimax lower bound (see [15], Theorem 2.5) states that for any norm where H is a standard Gaussian vector in R m .It is easy to check the proof and to show that this inequality remains true when • is a gauge.This gives us that and it is enough to observe that max |z|=1 z p,+ ≤ σ p (X).
Proof of Theorem 2. We may assume that p ≥ 1 since Theorem 2 is obviously true when 0 < p ≤ 1.Let m be the integer so that 1 ≤ p ≤ m < p + 1, thus m ≤ 2p.We use the notation of Lemma 4. We first condition on Γ.We have Otherwise by our assumption H(p, λ) there exists a gauge in R m such that From Lemma 3 we get We now take the expectation with respect to Γ and get where H is a standard m-dimensional Gaussian vector.The proof is concluded using Lemma 4 and the fact that γ −1 p √ p is bounded.Indeed, 5 Tail behavior of convex measures , where c is a universal constant.
Remark: Note that the parameter λ(p, r) in Theorem 5 is bounded by a universal constant if the parameters p and r are not close, for instance if r ≥ 2 max{1, p}.Theorem 6.Let r > 1 and let X be a (−1/r)-concave random vector in a finite dimensional Euclidean space.Then, for every 0 < p < r, where and c is a universal constant.
Proof.The proof may be reduced to the case of a centered random vector.Indeed, let X be a (−1/r)-concave random vector, then so is X − EX.Since , we may assume that X is centered.The theorem now follows immediately by Theorems 2 and 5.
Note that trivially a reverse inequality to ( 12) is valid, for p ≥ 1: Therefore Theorem 6 states an equivalence Since a log-concave measure is κ-concave for any κ < 0, we obtain Corollary 7.For any log-concave random vector X in a finite dimensional Euclidean space and any p > 0, where C > 0 is a universal constant.
As it was mentioned above, if X ∈ E is (−1/r)-concave then so is z, X for any z ∈ E. From Lemma 21, we have that for any 1 ≤ p < r, where C 1 (p, r) is defined in Lemma 21.Assume that r > 2. Let n be the dimension of E. If moreover X is centered and has the identity as the covariance matrix -such a vector is called an isotropic random vector -then one has for any z ∈ S n−1 and any 1 ≤ p < r, Since in that case, E|X| ≤ (E|X| 2 ) 1/2 = √ n, it follows from Theorem 6 that for any 1 ≤ p < r, Together with Markov's inequality this give the following Corollary.
Corollary 8. Let r > 2 and let X ∈ R n be a (−1/r)-concave isotropic random vector.Then for every t > 0, In particular, if r ≥ 2 √ n, then for every 6c ≤ t ≤ 3cr/ √ n, where c and c 0 are universal positive constants.
Remark.A log-concave measure is (−1/r)-concave for every r > 0, thus in such a case inequality ( 17) is valid for every t > c, which is a result from [26].
To prove the "In particular" part denote r ′ = t √ n/(3c).Note that r ′ ≥ 2 √ n and that r ′ ≤ r.Therefore X is (−1/r ′ )-concave as well and we can apply (16) with r ′ , obtaining the bound for probability 3 −r ′ /2 , which implies the result.
We now apply our results to the problem of the approximation of the covariance matrix by the empirical covariance matrix.Recall that for a random vector X the covariance matrix of X is given by EXX ⊤ .It is equal to the identity operator I if X is isotropic.The empirical covariance matrix of a sample of size N is defined by i , where X 1 , X 2 , . . ., X N are independent copies of X.The main question is how small N can be taken in order that these two matrices are close to each other in the operator norm (clearly, if X is non-degenerated then N ≥ n due to the dimensional restrictions and, by the law of large numbers, the empirical covariance matrix tends to the covariance matrix as N grows to infinity).See [1,2] for references on this question and for corresponding results in the case of log-concave measures.In particular, it was proved there that for N ≥ n and log-concave n-dimensional vectors , where, as usual, I is the identity operator, • is the operator norm ℓ n 2 → ℓ n 2 and c, C are absolute positive constants.
In [29] (Theorem 1.1), the following condition was introduced: an isotropic random vector X ∈ R n is said to satisfy the strong regularity assumption if for some η, C > 0 and every rank k ≤ n orthogonal projection P , one has for every We show that an isotropic (−1/r)-concave random vector satisfies this assumption.For simplicity we will show this with η = 1 (one can change η adjusting constants).Lemma 9. Let n ≥ 1, a > 0 and r = max{4, 2a log n}.Let X ∈ R n be an isotropic (−1/r)-concave random vector.Then there exists an absolute constant C such that for every rank k orthogonal projection P and every t ≥ C 1 (a), one has where C 1 (a) = C exp (4/a) and C 2 (a) = C max{(a log a) 4 , exp (32/a)}.
Proof.Let P be a projection of rank k.Let c be the constant from Corollary 8 (without loss of generality we assume c ≥ 1) and t > c.If r ≤ √ k then Corollary 8 implies that and applying Corollary 8 again we obtain Thus in both cases we have One can check that for t ≥ c 2 exp (4/a) and k ≥ exp (16/a) this implies which proves the desired result for k > C a := max{64a 2 log 2 (4a), exp (16/a)} and t ≥ c 2 exp (4/a).Assume now that k ≤ C a .Then we apply Borell's Lemma -Lemma 21.We have that for every t ≥ 3 .
It is not difficult to see (e.g., by considering cases t ≤ 9r, 9r < t ≤ 18r and t > 18r) that for C(a) := 54 4 C 2 a , t ≥ 3 and r ≥ 4, one has This completes the proof.
Theorem 1.1 from [29] and the above lemma immediately imply the following corollary on the approximation of the covariance matrix by the empirical covariance matrix.
Let r = 2a log(2n) > 8. Applying Corollary 8 for independent (−1/r)-concave isotropic random vectors X 1 , X 2 , . . .,X N and using results of [24], it can be checked that with large probability where C(a) depends only on a.As we mentioned above, this extends the results of [1,2] on the approximation of the covariance matrix from the log-concave setting to the class of convex measures.Now we prove Theorem 5. We need the following lemma.Recall that K α was defined by (5).
Lemma 11.Let m be an integer.Let r > 1 and 0 < p < r.Let Y ∈ R m be a centered random vector with density g = f −β with β = m + r and f convex positive.Let F : R m → R + be such that for every t ∈ R m , F (2t) ≤ 2 p F (t) and assume that EF (Y ) is finite.Then, there exists a positive universal constant c such that 0 ∈ K α (g) and where c(p, r) = 1 + c r−p and α = c (m+r) 2 (r−p)(r−1) and by definition, min f < α −m/(r+m) f (t) when t / ∈ K α (g).Using the convexity of f and the last two inequalities we get Let δ = δ(α) := 1 + γ −1 α −m/(r+m) r+m .The inequality (19) can be written Therefore and from the assumption on F , we get We conclude that if 2 p−r δ < 1 then , where c > 0 is a universal constant.This concludes the proof of (18).
Remark.An interesting setting for the previous lemma is when r is away from 1, for instance r ≥ 2, r and m are comparable, and p is proportional to r.In this case γ is bounded by a constant, c(p, r) explodes only when p → r, and α depends only on the ratio r/p.Proof of Theorem 5. Let 1 ≤ p < r and m = ⌈p⌉.Let A : E → R m be a linear mapping and Y = AX be a centered non-degenerated (−1/r)-concave random vector.By Borell's result [9,10], there exists a positive convex function f such that the distribution of Y has a density of the form g = f −(r+m) .
We apply Lemma 11 and use the notation of that lemma.Because the class of (−1/r)-concave measures increases as the parameter r decreases, we may assume that r ≤ 2p (note that λ(p, 2p) ∼ λ(p, r) for r > 2p, so we do not loose control of the constant assuming that r ≤ 2p).Thus 1 ≤ p ≤ m and r ≤ 2m.We deduce that the parameter α from Lemma 11 satisfies where c is a numerical constant.Now note that because g −1/(r+m) is convex, K = K α (g) is a convex body and from Lemma 11, it contains 0. Let • be its Minkowski functional.
We have so that and therefore Let F (t) = t p for t ∈ R m .Thus F (2t) = 2 p F (t) and, since p < r, EF (Y ) is finite.Hence F the assumption of Lemma 11.Therefore for c(p, r) = 1 + c/(r − p) Another application of Lemma 11, which will be used later, is the following lemma.
Lemma 12. Let 1 ≤ p < r and m = ⌈p⌉.Let Y ∈ R m be a centered (−1/r)concave random vector with density g.There exists a universal constant c, such that 0 ∈ K α (g) and where , and c > 0 is a universal constant.
Proof.Repeating the above proof with the function F (t) = | x, t | p we obtain that 0 ∈ K α (g) and which implies the result.

Small ball probability estimates
The following result was proved in [27].
Theorem 13.Let X be a centered log-concave random vector in a finite dimensional Euclidean space.For every ε (0, c ′ ) one has where c, c ′ > 0 are universal positive constants.
In this section we generalize this result to the setting of convex distributions.We first establish a lower bound for the negative moment of the Euclidean norm of a convex random vector.Theorem 14.Let r > 1 and let X be a centered n-dimensional (−1/r)-concave random vector.Assume 1 ≤ p < min{r, n/2}.Then where and c, C are absolute positive constants.Moreover, if 0 < p < 1 then where c 0 is an absolute positive constant.
From Markov's inequality we deduce a small ball probability estimates for convex measures.
We need the following result from [18] (Theorem 1.3 there).
Theorem 16.Let n ≥ 1 be an integer, • be a norm in R n and K be its unit ball.Assume that 0 < p ≤ c 0 (E G /σ) 2 and m = ⌈p⌉.Then where µ = µ n,m and c is an absolute positive constant.
The proof of Theorem 14 is based on the following two lemmas.
Lemma 17.Let m ≤ n, α > 0 and X be a random vector in R n with density g.Then, Proof.Integrating in polar coordinates (see [27], Proposition 4.6), we obtain the following key formula This implies the result, since g F (0) ≤ g F ∞ .
Below we will use the following notation.For a random vector X in R n , p > 0, and t ∈ R n we denote (note that it is the dual gauge of the so-called centroid body, which is rather an L p -norm than the ℓ p -norm).
Lemma 18.Let 1 ≤ p < r and m = ⌈p⌉.Let X be a centered (−1/r)-concave random vector in R n with density g.Let K denote the unit ball of • p .Then for every m-dimensional subspace F ⊂ R n one has , and c > 0 is a universal constant.
Proof.Applying Lemma 12 to Y = P F X, we obtain that for every t ∈ F with α and C 3 (p, r) given in Lemma 12. Since for t ∈ F , ||t|| p = max x, t , where the supremum is taken over x ∈ (K ∩ F ) • = P F K • , this is equivalent to By Rogers-Sheppard inequality [28] we observe This implies the result.
Proof of Theorem 14.
Recall that c 1 , c 2 , ... denote absolute positive constants.Recall also that for a random vector X in R n , p > 0, and t ∈ R n t p = (E| X, t | p ) 1/p .
Finally, let K denote the unit ball of • p .We assume that X is non-degenerate in R n and let m = ⌈p⌉.Without loss of generality we assume that where C is a large enough absolute constant.
As in (10), since p ≤ m ≤ 2p, we have Hence This implies that for sufficiently large C we have m ≤ 2p ≤ c 0 (E G p /σ p (X)) 2 , where c 0 is the constant from Theorem 16.
Note that (E|G| −p ) −1/p ≥ (E|G| −m ) −1/m ≥ c 3 √ n (the second inequality is well known for m ≤ n/2 and can be directly computed).Combining Lemmas 17 and 18, we obtain , with α and C 3 (p, r) in Lemma 18.Now note that Applying the first inequality from (21), we obtain the desired result.
The "Moreover" part is an immediate corollary of Lemmas 22 (with q = 1) and 23.
Conjecture.We conjecture that for convex distributions a similar thin shell property holds as for log-concave distribution: if X is an isotropic (−1/r)concave random vector in R n with r > 2, then ∀t ∈ (0, 1) as n tends to ∞. See [17] for recent work in the log-concave setting.

Appendix
There is a vast literature on inequalities of integrals related to concave functions.Some of the following lemmas may be known but we did not find any reference.Their proof use classical methods for demonstrating integral inequalities involving concave functions (see [11] and [25]).The first lemma is a mirror image for negative moments of a result from [23] valid for positive moments.
Lemma 19.Let s, m, β ∈ R such that β > m + 1 > 0 and s > 0. Let ϕ be a non-negative concave function on [s, +∞).Then where a is chosen so that H(∞) = 0. Note that H ′ , the derivative of H, has the same sign as (ϕ(x)/(x − s)) m − a m .Since ϕ(x)/(x − s) is decreasing on (s, +∞), we deduce that H is first increasing and then decreasing.Since H(s) = H(∞) = 0 we conclude that H is non-negative.This means that for every t ≥ s, Now, note that for any β ′ > β and any non-negative function F , we have by Fubini's theorem, Using (22) and applying this relation to F = ϕ m and then to F (x) = a m (x−s) , we get that From the definition of a, we conclude that . Lemma 20.Let m ≥ 1 be an integer.Let g be the density of a probability on R m of the form g = f −β with f positive convex on R m and β > m + 1.If xg(x) dx = 0, then Proof.Since f is convex it follows from Jensen's inequality that where s = min f = g The conclusion follows from combining the last two inequalities.

Remark.
When β → ∞, which corresponds to a log-concave density, we recover the inequality from [13] saying that g(0) ≥ e −m g ∞ .
The next lemma is a well known result of Borell ([9]) stated in a way that fits our needs and stresses the dependence on the parameter of concavity.Proof.Denote θ := P ( X ≤ 3E X ).Assume that θ < 1 (otherwise we are done).From Markov's inequality, θ = 1 − P( X > 3E X ) ≥ 2/3.
Integrating, we get The following, a stronger variant of Borell's lemma, allows to compare the expectation of a random variable X and a median Med( X ).It was proved in [7] (Theorem 1.1, see also the discussion following Theorem 5.2 for the behavior of the corresponding constant).It was also implicitly proved in [16] (see inequality (4) in [14]).The second part of the lemma follows by integration.Lemma 22.Let r > 1 and X be a (−1/r)-concave random vector in R m .Then for any semi-norm • and any t ≥ 1, one has P ( X ≥ tMed( X )) ≤ (C 0 r) r t −r , where C 0 is an absolute positive constant.As a consequence, for every r > q ≥ 1 one has (E X q ) 1/q ≤ Cr r r − q 1/q Med( X ), where C is an absolute positive constant.
The following lemma is Corollary 9 from [14] (as before, the second part follows by integration).Lemma 23.Let r > 1 and X be a (−1/r)-concave random vector in R m .Then for any semi-norm • and any ε ∈ (0, 1), one has P ( X ≤ εMed( X )) ≤ C 0 ε, where C 0 is an absolute positive constant.As a consequence, for every p ∈ (0, 1), E X −p −1/p ≥ c(1 − p)Med( X ), where c is an absolute positive constant.

Theorem 5 .
Let n ≥ 1 and r > 1.Let X be a centered (−1/r)-concave random vector in a finite dimensional Euclidean space.Then for every 0 < p < r, X satisfies the assumption H(p, λ(p, r)) with λ(p, r) = c r r−1 3 r r−p 4