Vector-Valued Maclaurin Inequalities
aa r X i v : . [ m a t h . M G ] F e b Vector-Valued Maclaurin Inequalities
Silouanos Brazitikos, Finlay McIntyreFebruary 12, 2021
Abstract
We investigate a Maclaurin inequality for vectors and its connection to an Aleksandrov-typeinequality for parallelepipeds.
The classical Maclaurin inequality compares consecutive symmetric sums for any sequence of positivereal numbers:
Theorem 1 (Maclaurin inequality) . For any sequence of positive real numbers x , . . . , x m and any ă k ď m , the following inequality holds: ¨˝ ř ď i 㨨¨ă i k ď m x i ¨ ¨ ¨ x i k ` mk ˘ ˛‚ k ď ¨˚˝ ř ď i 㨨¨ă i k ´ ď m x i ¨ ¨ ¨ x i k ´ ` mk ´ ˘ ˛‹‚ k ´ . This was first proved in [15] and a proof of this result using elementary methods can be found also in[8]. The Maclaurin inequality can be seen as a refinement of the arithmetic-geometric mean inequalityby noting that the geometric mean and arithmetic mean appear as the smallest and largest quantitiesrespectively in the following chain of inequalities: p x ¨ ¨ ¨ x m q m ď ¨˚˝ ř ď i 㨨¨ă i m ´ ď m x i ¨ ¨ ¨ x i m ´ ` mm ´ ˘ ˛‹‚ m ´ ď ¨ ¨ ¨ ď ¨˚˝ ř ď i ă j ď m x i x j ` m ˘ ˛‹‚ ď ř mi “ x i m , which follows directly from Theorem 1.In this note, we explore a variant of this classical result, with sequences of numbers replaced byfamilies of vectors, and the standard product replaced by the wedge product operator. More precisely,let v , . . . , v m P R d with d ď m , and for any ď i ă ¨ ¨ ¨ ă i k ď m with k ď d denote by | v i ^¨ ¨ ¨^ v i k | the k -dimensional volume of the parallelotope spanned by v i , . . . , v i k . We are interested in "vector-valued"inequalities of the form ¨˝ ř ď i 㨨¨ă i k ď m | v i ^ ¨ ¨ ¨ ^ v i k | p ` mk ˘ ˛‚ kp ď ¨˚˝ ř ď i 㨨¨ă i k ´ ď m | v i ^ ¨ ¨ ¨ ^ v i k ´ | p ` mk ´ ˘ ˛‹‚ p k ´ q p , (1)with p P r , and ď k ď d . Note that if m “ d and v , . . . , v m are orthogonal, then each term | v i ^ ¨ ¨ ¨ ^ v i k | p will just be equal to a k -fold product of numbers, namely } v i } p ¨ ¨ ¨ } v i k } p . It followsthat (1) reduces to a special case of the classical Maclaurin inequality for any p P p , .Given a general family of vectors v , . . . , v m P R d , the value of p plays a more important role. Usingelementary results from linear algebra, we are able to establish (1) for p “ and m “ d :1 heorem 2. For any d -tuple of vectors v , . . . , v d P R d and any ă k ď d , the following inequalityholds: ¨˚˝ ř ď i 㨨¨ă i k ď d | v i ^ ¨ ¨ ¨ ^ v i k | ` dk ˘ ˛‹‚ k ď ¨˚˝ ř ď i 㨨¨ă i k ´ ď d | v i ^ ¨ ¨ ¨ ^ v i k ´ | ` dk ´ ˘ ˛‹‚ k ´ . Moreover, taking limits as p Ñ 8 , we can write lim p Ñ8 ¨˝ ř ď i 㨨¨ă i k ď m | v i ^ ¨ ¨ ¨ ^ v i k | p ` mk ˘ ˛‚ p “ max ď i 㨨¨ă i k ď m | v i ^ ¨ ¨ ¨ ^ v i k | . Again using purely linear algebra, namely Szasz’s inequality for subdeterminants, we are able to provethe following endpoint case:
Theorem 3.
Fix vectors v , . . . , v m P R d with ď d ď m . Then, for any ă k ď d , the followinginequality holds: ˆ max ď i 㨨¨ă i k ď m | v i ^ ¨ ¨ ¨ ^ v i k | ˙ k ď ˆ max ď i 㨨¨ă i k ´ ď m | v i ^ ¨ ¨ ¨ ^ v i k ´ | ˙ k ´ . (2)By a similar argument, we also prove (1) for p “ : Theorem 4.
Fix vectors v , . . . , v m P R d with ď d ď m . Then, for any ă k ď d , the followinginequality holds: ˜ ź ď i 㨨¨ă i k ď m | v i ^ ¨ ¨ ¨ ^ v i k | ¸ p mk q k ď ¨˝ ź ď i 㨨¨ă i k ´ ď m | v i ^ ¨ ¨ ¨ ^ v i k ´ | ˛‚ p mk ´ q p k ´ q . It is not difficult to verify that (1) fails to hold in general for negative values of p .If we take p “ , it seems more difficult to establish the desired inequality for a general family ofvectors. However, in the case where m “ d , we have some partial results: Theorem 5. If m “ d , then for any v , . . . , v m P R d , inequality (1) holds with p “ and k “ , , d inall dimensions d . Using a certain duality between families of vectors, one can also prove the case for p “ and k “ in dimensions and - see [13], Section 6 for details. To prove Theorem 5, in each case we essentiallyconstruct a new family of orthogonal vectors ˜ v , . . . , ˜ v m P R d such that ¨˝ ř ď i 㨨¨ă i k ď m | v i ^ ¨ ¨ ¨ ^ v i k | ` mk ˘ ˛‚ k ď ¨˝ ř ď i 㨨¨ă i k ď m | ˜ v i ^ ¨ ¨ ¨ ^ ˜ v i k | ` mk ˘ ˛‚ k , and ¨˚˝ ř ď i 㨨¨ă i k ´ ď m | v i ^ ¨ ¨ ¨ ^ v i k ´ | ` mk ´ ˘ ˛‹‚ k ´ ě ¨˚˝ ř ď i 㨨¨ă i k ´ ď m | ˜ v i ^ ¨ ¨ ¨ ^ ˜ v i k ´ | ` mk ´ ˘ ˛‹‚ k ´ . The desired result then follows by a simple application of Theorem 1.In light of these results, we conjecture that the vector-valued Maclaurin inequalities should hold inthe full range ď p ď 8 : 2 onjecture 1 (Vector-valued Maclaurin inequality) . Fix vectors v , . . . , v m P R d with ď d ď m . Thenfor all p P r , and ď k ď d , the following inequality holds: ¨˝ ř ď i 㨨¨ă i k ď m | v i ^ ¨ ¨ ¨ ^ v i k | p ` mk ˘ ˛‚ kp ď ¨˚˝ ř ď i 㨨¨ă i k ´ ď m | v i ^ ¨ ¨ ¨ ^ v i k ´ | p ` mk ´ ˘ ˛‹‚ p k ´ q p , with equality if and only if m “ d and the vectors v i form an orthonormal basis. It should be noted that for p “ , the vector-valued Maclaurin inequality is of particular interest, as itturns out to be closely related to the far-reaching Aleksandrov–Fenchel inequality from convex geometry.By a simple argument one can deduce the classical Maclaurin inequality as a consequence of Newton’sinequality, and similarly one would be able to deduce the vector-valued Maclaurin inequality for p “ from a corresponding vector-valued version of Newton’s inequality of the following form ¨˝ ř ď i 㨨¨ă i k ď m | v i ^ ¨ ¨ ¨ ^ v i k | ` mk ˘ ˛‚ ě ¨˚˝ ř ď i 㨨¨ă i k ´ ď m | v i ^ ¨ ¨ ¨ ^ v i k ´ | ` mk ´ ˘ ˛‹‚¨˚˝ ř ď i 㨨¨ă i k ` ď m | v i ^ ¨ ¨ ¨ ^ v i k ` | ` mk ` ˘ ˛‹‚ , (3)where ď k ď d ´ . It is worth noting the explicit connection between Newton’s inequality and theAleksandrov–Fenchel inequality. Up until this point, in the literature, the Aleksandrov–Fenchel inequalityhas been referred to as a Newton-type inequality, simply because it is of the same form (square greaterthan a product).To illustrate the connection between the vector-valued Maclaurin inequality and the Aleksandrov–Fenchel inequality, let us denote by P the following Minkowski sum of line segments P “ m ÿ j “ r´ v j , v j s . Using this notation, Conjecture 1 with p “ exactly states that for ă k ď d we have ˜ V k p P q ` mk ˘ ¸ k ď ˜ V k ´ p P q ` mk ´ ˘ ¸ k ´ , (4)or ˆ V k p P q V k p C m q ˙ k ď ˆ V k ´ p P q V k ´ p C m q ˙ k ´ , where V k denotes the k -th intrinsic volume and C m is the m -dimensional unit cube. This is a generalisoperimetric-type inequality. For example, the case m “ d and and k “ d , which is proved here, saysthat among all parallelepipeds (non necessarily orthogonal) with the same volume, the cube has thesmallest surface area. This particular case was first proved by Hadwiger in [7] (see also [6]) using Steinersymmetrisation. Moreover, if (4) holds for an arbitrary sum of line segments, then one would recover thedimension free estimate of McMullen [11] restricted to the class of zonoids. Furthermore, (4) is relatedto isoperimetric-type inequalities proved in [9].Despite the fact that we don’t have a proof for the sharp inequality (4), we are able to prove it witha constant that it is bounded by an absolute constant that doesn’t depend on the dimension. Theorem 6.
For any d -tuple of vectors v , . . . , v d P R d and any ă k ď d , the following inequalityholds: ¨˚˝ ř ď i 㨨¨ă i k ď d | v i ^ ¨ ¨ ¨ ^ v i k | ` dk ˘ ˛‹‚ k ď p d ´ k ` qp d ´ k ` q ¨˚˝ ř ď i 㨨¨ă i k ´ ď d | v i ^ ¨ ¨ ¨ ^ v i k ´ | ` dk ´ ˘ ˛‹‚ k ´ Note that the constant appearing on the right-hand side is greater than , but smaller than 2.3 tructure of paper In Section 2 we introduce some relevant notation and terminology. Section 3 is dedicated to provingTheorems 2, 3 and 4. In Section 4, we discuss a general approach for attempting to establish (1) with p “ , and prove the special cases listed in Theorem 5. At the beginning of Section 5, we introduce toolsfrom convex geometry which allow us to rewrite the vector-valued Maclaurin inequality with p “ interms of convex bodies and mixed volumes. In Section 5.1, we then use these tools to establish Theorem6. We work in R d , which is equipped with a Euclidean structure x¨ , ¨y and we fix an orthonormal basis t e , . . . , e d u . We denote by B d and S d ´ the Euclidean unit ball and sphere in R d respectively. We write σ for the normalised rotationally invariant probability measure on S d ´ and ν for the Haar probabilitymeasure on the orthogonal group O p d q . Let G d,k denote the Grassmannian of all k -dimensional subspacesof R d . Then, O p d q equips G d,k with a Haar probability measure ν d,k . The letters c, c , c , c etc. denoteabsolute positive constants which may change from line to line. Whenever we write a » b , we mean thatthere exist absolute constants c , c ą such that c a ď b ď c a .Let K d denote the class of all non-empty compact convex subsets of R d . If K P K d has non-empty interior, we will say that K is a convex body. If A P K d , we will denote by | A | the volumeof A in the appropriate affine subspace unless otherwise stated. The volume of B d is denoted by ω d .We say that a convex body K in R d is symmetric if x P K implies that ´ x P K , and that K iscentred if its centre of mass | K | ş K x dx is at the origin. The support function of a convex body K isdefined by h K p y q “ max tx x, y y : x P K u . For any E P G d,k we denote by E K the orthogonal subspaceof E , i.e. E K “ t x P R d : x x, y y “ for all y P E u . In particular, for any u P S d ´ we define u K “ t x P R d : x x, u y “ u . The section of K P K d with a subspace E of R d is K X E , and the orthogonalprojection of K onto E is denoted by P E p K q .Mixed volumes are introduced by a classical theorem of Minkowski which describes the way volumebehaves with respect to the operations of addition and multiplication of compact convex sets by non-negative reals: if K , . . . , K N P K d , N P N , then the volume of t K ` ¨ ¨ ¨ ` t N K N is a homogeneouspolynomial of degree d in t i ě (see [3] and [16]): ˇˇ t K ` ¨ ¨ ¨ ` t N K N ˇˇ “ ÿ ď i ,...,i d ď N V p K i , . . . , K i d q t i . . . t i d , (5)where the coefficients V p K i , . . . , K i d q are chosen to be invariant under permutations of their arguments.The coefficient V p K i , . . . , K i d q is called the mixed volume of the d -tuple p K i , . . . , K i d q . We will oftenuse the fact that V is positive linear with respect to each of its arguments and that V p K, . . . , K q “ | K | (the d -dimensional Lebesgue measure of K ) for all K P K d .Steiner’s formula is a special case of Minkowski’s theorem. If K P K d then the volume of K ` tB d , t ą , can be expanded as a polynomial in t : | K ` tB d | “ d ÿ k “ ˆ dk ˙ W k p K q t k , (6)where W k p K q : “ V p K r d ´ k s , B d r k sq is the k -th quermassintegral of K . Moreover, for k “ , . . . , d , the k -th intrinsic volume of a convex body L Ă R d is defined as V k p L q “ ˆ dk ˙ V p L r k s , B d r d ´ k sq ω d ´ k . The Aleksandrov-Fenchel inequality states that if
K, L, K , . . . , K d P K d , then V p K, L, K , . . . , K d q ě V p K, K, K , . . . , K d q V p L, L, K , . . . , K d q . (7)In particular, this implies that the sequence p W p K q , . . . , W d p K qq is log-concave. From the Aleksandrov-Fenchel inequality one can recover the Brunn-Minkowski inequality as well as the following generalisation4or the quermassintegrals: W k p K ` L q d ´ k ě W k p K q d ´ k ` W k p L q d ´ k , k “ , . . . , d ´ . (8)We write S p K q for the surface area of K . From Steiner’s formula and the definition of surface area wesee that S p K q “ dW p K q . Finally, let us mention Kubota’s integral formula W k p K q “ ω d ω d ´ k ż G d,d ´ k | P E p K q| dν d,d ´ k p E q , ď k ď d ´ . (9)The case k “ is Cauchy’s surface area formula S p K q “ ω d dω d ´ ż S d ´ | P u K p K q| dσ p u q . (10)We refer to the books [4] and [16] for basic facts from the Brunn-Minkowski theory and to the books [1]and [2] for basic facts from asymptotic convex geometry. In this section we introduce some elementary tools from linear algebra, and use them to prove Theorems2, 3 and 4.
Vector-valued Maclaurin inequality with p “ Firstly, for any family of vectors v . . . , v d P R d ,we write the square of each k -dimensional volume | v i ^ ¨ ¨ ¨ v i k | as the determinant of a k ˆ k submatrixof some fixed d ˆ d matrix. Let us introduce the notion of a principal minor: Definition 1 (Principal minors) . Let M be an n ˆ n matrix. For S Ă r n s , define M S to be submatrixconstructed by removing rows and edges with indices not in S . The set of principal submatrices of M isdefined as t M S | S Ă r n su . Furthermore, the set of principal minors of M is defined as t det p M S q | S Ăr n su . For ď k ď n , we define the principal k -submatrices and principal k -minors of M by adding thecondition that | S | “ k .Let A denote the square d ˆ d matrix with columns v , . . . , v d and let B be the d ˆ k matrix with columns v , . . . , v k for some ď k ď d , then we can write | v ^ ¨ ¨ ¨ ^ v k | “ det p B T B q . A short proof of this identity can be found in [10]. The matrix B T B is a principal k -submatrix of A T A ,and can be constructed by removing the last p d ´ k q rows and columns. In this way we see that the sumof terms | v i ^ ¨ ¨ ¨ ^ v i k | over all ď i ă ¨ ¨ ¨ ă i k ď d is equal to the sum of all principal k -minors of A T A . The next lemma allows us to work with the sum of all principal minors of A T A : Lemma 1 (Sum of principal minors) . Let M be a n ˆ n matrix with eigenvalues λ , . . . , λ n (not necessarilydistinct). For ď k ď n we have that ÿ | S |“ k det p M TS M S q “ ÿ | S |“ k ź i P S λ i . A proof of this lemma can be found in [12]. Now we are ready to establish the vector-valued Maclaurininequality with p “ : Proof of Theorem 2.
As before, let A denote the square matrix with columns v , . . . , v d and (not neces-sarily distinct) eigenvalues λ , . . . , λ d . Then for ď i ă ¨ ¨ ¨ ă i k ď d , define B i ,...,i k to be the d ˆ k v i , . . . , v i k . Now we use Lemma 1 along with Theorem 1, which is precisely theclassical Maclaurin inequality, to write ¨˚˝ ř ď i 㨨¨ă i k ď d | v i ^ ¨ ¨ ¨ ^ v i k | ` dk ˘ ˛‹‚ k “ ¨˚˝ ř ď i 㨨¨ă i k ď d det ` B Ti ,...,i k B i ,...,i k ˘` dk ˘ ˛‹‚ k “ ¨˚˝ ř ď i 㨨¨ă i k ď d λ i ¨ ¨ ¨ λ i k ` dk ˘ ˛‹‚ k ď ¨˚˝ ř ď i 㨨¨ă i k ´ ď d λ i ¨ ¨ ¨ λ i k ´ ` dk ´ ˘ ˛‹‚ k ´ “ ¨˚˚˝ ř ď i 㨨¨ă i k ´ ď d det ´ B Ti ,...,i k ´ B i ,...,i k ´ ¯` dk ´ ˘ ˛‹‹‚ k ´ “ ¨˚˝ ř ď i 㨨¨ă i k ´ ď d | v i ^ ¨ ¨ ¨ ^ v i k ´ | ` dk ´ ˘ ˛‹‚ k ´ . This proves the desired result.
Endpoint cases p “ and p “ 8 To begin, let us state a result of Szasz regarding principal minors:
Lemma 2 (Szasz’s inequality) . Let M be some n ˆ n matrix. For ă k ă n the following inequalityholds ¨˝ ź | A |“ k det p M A q ˛‚ p n ´ k ´ q ď ¨˝ ź | B |“ k ´ det p M B q ˛‚ p n ´ k ´ q . A proof of this using elementary methods can be found in [10]. Simple applications of this lemmaallow us to deduce the vector-valued Maclaurin inequality with p “ and p “ 8 . Proof of Theorem 4.
Let A denote the d ˆ m matrix with columns v , . . . , v m and let A i ,...,i k be the d ˆ k matrix with colums v i , . . . , v i k for some ď k ď d , then we have | v i ^ ¨ ¨ ¨ ^ v i k | “ det p A Ti ,...,i k A i ,...,i k q . As we mentioned earlier, A Ti ,...,i k A i ,...,i k can also be seen as a principal submatrix of A T A , which is an m ˆ m matrix. Hence, by Szasz’s lemma we have ˜ ź ď i 㨨¨ă i k ď m | v i ^ ¨ ¨ ¨ ^ v i k | ¸ p m ´ k ´ q ď ¨˝ ź ď i 㨨¨ă i k ´ ď m | v i ^ ¨ ¨ ¨ ^ v i k ´ | ˛‚ p m ´ k ´ q . Taking square roots, and noting that ` m ´ k ´ ˘ “ ` mk ˘ ¨ km and ` m ´ k ´ ˘ “ ` mk ´ ˘ ¨ k ´ m , the above inequalitycan be written as ¨˝˜ ź ď i 㨨¨ă i k ď m | v i ^ ¨ ¨ ¨ ^ v i k | ¸ p mk q ˛‚ k ď ¨˚˝¨˝ ź ď i 㨨¨ă i k ´ ď m | v i ^ ¨ ¨ ¨ ^ v i k ´ | ˛‚ p mk ´ q ˛‹‚ k ´ , (11)as required. 6 emark . In (11) we see that we have geometric means appearing inside the first set of parenthesis onboth sides. Intriguingly, simply replacing these geometric means by arithmetic means reveals the vectorvalued Maclaurin inequality with p “ . So, if we think of the vector-valued Maclaurin inequality with p “ as a chain of inequalities for a sequence of arithmetic means, then (11) can be thought of as ananalogous chain of inequalities for the corresponding geometric means.The case for p “ 8 also follows directly from Szasz’s inequality. Proof of Theorem 3.
Let i ă ¨ ¨ ¨ ă i k ď m be the indices where the left hand side is maximised and let B be the matrix with columns v i , . . . , v i k . Now, using Szasz’s inequality for M “ B T B and n “ k weget | v i ^ ¨ ¨ ¨ ^ v i k | ď ˜ k ź j “ | v i ^ ¨ ¨ ¨ ^ x v i j ^ ¨ ¨ ¨ ^ v i k | ¸ k ´ ď ˜ k ź j “ ˆ max ď i 㨨¨ă i k ´ ď d | v i ^ ¨ ¨ ¨ ^ v i k ´ | ˙¸ k ´ “ ˆ max ď i 㨨¨ă i k ´ ď d | v i ^ ¨ ¨ ¨ ^ v i k ´ | ˙ kk ´ , which concludes the proof. p “ and monotonicity argument In the next section we prove the special cases of vector-valued Maclaurin inequalities with p “ and m “ d listed in Theorem 5. Our method is somewhat inspired by a monotonicity argument given in [8]to prove the classical Maclaurin inequality. Given vectors v , . . . , v d P R d , we attempt to construct asecond family of orthogonal vectors ˜ v , . . . , ˜ v d P R d , such that ¨˚˝ ř ď i 㨨¨ă i k ď d | v i ^ ¨ ¨ ¨ ^ v i k | ` dk ˘ ˛‹‚ k ď ¨˚˝ ř ď i 㨨¨ă i k ď d | ˜ v i ^ ¨ ¨ ¨ ^ ˜ v i k | ` dk ˘ ˛‹‚ k , and ¨˚˝ ř ď i 㨨¨ă i k ´ ď d | v i ^ ¨ ¨ ¨ ^ v i k ´ | ` dk ´ ˘ ˛‹‚ k ´ ě ¨˚˝ ř ď i 㨨¨ă i k ´ ď d | ˜ v i ^ ¨ ¨ ¨ ^ ˜ v i k ´ | ` dk ´ ˘ ˛‹‚ k ´ .
7f such an orthogonal family exists, then applying Theorem 1 with positive numbers } v } , . . . , } v d } , wecan write ¨˚˝ ř ď i 㨨¨ă i k ď d | v i ^ ¨ ¨ ¨ ^ v i k | ` dk ˘ ˛‹‚ k ď ¨˚˝ ř ď i 㨨¨ă i k ď d | ˜ v i ^ ¨ ¨ ¨ ^ ˜ v i k | ` dk ˘ ˛‹‚ k “ ¨˚˝ ř ď i 㨨¨ă i k ď d } ˜ v i } ¨ ¨ ¨ } ˜ v i k } ` dk ˘ ˛‹‚ k ď ¨˚˝ ř ď i 㨨¨ă i k ´ ď d } ˜ v i } ¨ ¨ ¨ } ˜ v i k ´ } ` dk ´ ˘ ˛‹‚ k ´ “ ¨˚˝ ř ď i 㨨¨ă i k ´ ď d | ˜ v i ^ ¨ ¨ ¨ ^ ˜ v i k ´ | ` dk ´ ˘ ˛‹‚ k ´ ď ¨˚˝ ř ď i 㨨¨ă i k ´ ď d | v i ^ ¨ ¨ ¨ ^ v i k ´ | ` dk ´ ˘ ˛‹‚ k ´ . In order to streamline the argument slightly, we introduce the following convenient notation S k p v , . . . , v d q : “ ÿ t i ,...,i k uĂr d s | v i ^ ¨ ¨ ¨ ^ v i k | . Notice that by symmetry, instead of constructing a whole family of orthogonal vectors, it suffices toconstruct ˜ v P t v , . . . , v d u K such that S k p v , . . . , v d q ď S k p ˜ v , v , . . . , v d q , and S k ´ p v , . . . , v d q ě S k ´ p ˜ v , v , . . . , v d q . Since we only need to worry about the terms involving v , these last two conditions can be writtenrespectively as ÿ t i ,...,i k ´ uĂr d sz | v ^ v i ^ ¨ ¨ ¨ ^ v i k ´ | ď ÿ t i ,...,i k ´ uĂr d sz | ˜ v ^ v i ^ ¨ ¨ ¨ ^ v i k ´ | , and ÿ t i ,...,i k ´ uĂr d sz | v ^ v i ^ ¨ ¨ ¨ ^ v i k ´ | ě ÿ t i ,...,i k ´ uĂr d sz | ˜ v ^ v i ^ ¨ ¨ ¨ ^ v i k ´ | . Note that ˜ v is orthogonal to the vectors v , . . . , v d , so for any t v i , . . . , v i r u Ă t v , . . . , v d u| ˜ v ^ v i ^ ¨ ¨ ¨ ^ v i r | “ } ˜ v }| v i ^ ¨ ¨ ¨ ^ v i r | . Hence we can rewrite the previous two inequalities as follows, ř t i ,...,i k ´ uĂr d sz | v ^ v i ^ ¨ ¨ ¨ ^ v i k ´ | ř t i ,...,i k ´ uĂr d sz | v i ^ ¨ ¨ ¨ ^ v i k ´ | ď } ˜ v } ď ř t i ,...,i k ´ uĂr d sz | v ^ v i ^ ¨ ¨ ¨ ^ v i k ´ | ř t i ,...,i k ´ uĂr d sz | v i ^ ¨ ¨ ¨ ^ v i k ´ | . So the question is can choose a length } ˜ v } satisfying the above? Let us summarise what we have justderived with the following result: 8 heorem 7. Fix ă k ď d . Suppose that for any v , . . . , v d P R d ř t i ,...,i k ´ uĂr d sz | v ^ v i ^ ¨ ¨ ¨ ^ v i k ´ | ř t i ,...,i k ´ uĂr d sz | v i ^ ¨ ¨ ¨ ^ v i k ´ | ď ř t i ,...,i k ´ uĂr d sz | v ^ v i ^ ¨ ¨ ¨ ^ v i k ´ | ř t i ,...,i k ´ uĂr d sz | v i ^ ¨ ¨ ¨ ^ v i k ´ | , (12) where we interpret the k “ case as ř i Pr d sz | v ^ v i | ř i Pr d sz } v i } ď } v } . Then, for any ω , . . . , ω d P R d , we have ˜ S k p ω , . . . , ω d q ` dk ˘ ¸ k ď ˜ S k ´ p ω , . . . , ω d q ` dk ´ ˘ ¸ k ´ . Now we will prove various special cases of (12) which then imply the corresponding cases listed inTheorem 5:
Lemma 3. If k “ or k “ d , then p q holds for arbitrary v , . . . , v d P R d .Proof. To begin, let us deal with the case where k “ d . Observe that for ď i ď d we have | v ^ ¨ ¨ ¨ ^ v d || v ^ ¨ ¨ ¨ ^ v d | “ } π t v ,...,v d u K v } ď } π t v ,..., x v i ,...,v d u K v } “ | v ^ ¨ ¨ ¨ ^ p v i ^ ¨ ¨ ¨ ^ v d || v ^ ¨ ¨ ¨ ^ p v i ^ ¨ ¨ ¨ ^ v d | . Rearranging this gives | v ^ ¨ ¨ ¨ ^ v d || v ^ ¨ ¨ ¨ ^ p v i ^ ¨ ¨ ¨ ^ v d | ď | v ^ ¨ ¨ ¨ ^ p v i ^ ¨ ¨ ¨ ^ v d || v ^ ¨ ¨ ¨ ^ v d | . Summing over i yields d ÿ i “ | v ^ ¨ ¨ ¨ ^ v d || v ^ ¨ ¨ ¨ ^ p v i ^ ¨ ¨ ¨ ^ v d | ď d ÿ i “ | v ^ ¨ ¨ ¨ ^ p v i ^ ¨ ¨ ¨ ^ v d || v ^ ¨ ¨ ¨ ^ v d | which implies that | v ^ ¨ ¨ ¨ ^ v d || v ^ ¨ ¨ ¨ ^ v d | ď ř di “ | v ^ ¨ ¨ ¨ ^ p v i ^ ¨ ¨ ¨ ^ v d | ř di “ | v ^ ¨ ¨ ¨ ^ p v i ^ ¨ ¨ ¨ ^ v d | , which is exactly p q for k “ d . Now for the case where k “ . Simply note that d ÿ i “ | v ^ v i | ď } v } d ÿ i “ } v i } , which immediately gives ř di “ | v ^ v i | ř di “ } v i } ď } v } , which concludes the proof.Next we deal with the case for k “ , which requires a little more work: Lemma 4. If k “ , then p q holds in all dimensions d .
9n order to prove Lemma 4, we first prove the following variant of (12):
Lemma 5. ř ă i 㨨¨ă i d ´ ď d | v ^ v i ^ ¨ ¨ ¨ ^ v i d ´ | ř ă i 㨨¨ă i d ´ ď d | v i ^ ¨ ¨ ¨ ^ v i d ´ | ď ř ă i ď d | v ^ v i | ř ă i ď d } v i } . (13)Of course when d “ , (13) is exactly (12) with k “ .To prove Lemma 5 we need use an elementary fact regarding barycentric coordinates with respect asimplex. The next result is originally to Möbius [14]: Proposition 1 (Barycentric coordinates with respect to a simplex) . Let v , . . . , v d P R d ´ be the verticesof a p d ´ q -simplex ∆ . Given a vector u P ∆ , there exists a unique d -tuple p β , . . . , β d q P R d with ř dj “ β j “ , satisfying the following identity d ÿ j “ β j v j “ u. Furthermore, we can write such numbers β , . . . , β d explicitly using the following formula β j “ |p v ´ u q ^ ¨ ¨ ¨ ^ p v j ´ ´ u q ^ p v j ` ´ u q ^ ¨ ¨ ¨ ^ p v d ´ u q| p ∆ q . Proof of Lemma 5.
By rearranging it suffices to prove ¨˝ ÿ ă i 㨨¨ă i d ´ ď d | v ^ v i ^ ¨ ¨ ¨ ^ v i d ´ | ˛‚˜ ÿ ă i ď d } v i } ¸ ď ˜ ÿ ă i ď d | v ^ v i | ¸ ¨ ¨˝ ÿ ă i 㨨¨ă i d ´ ď d | v i ^ ¨ ¨ ¨ ^ v i d ´ | ˛‚ . For j “ , . . . , d ´ , it is always true that | v ^ v i ^ ¨ ¨ ¨ ^ v i d ´ || v i ^ ¨ ¨ ¨ ^ v i d ´ | “ ››› π t v i ,...,v id ´ u K v ››› ď ››› π v ij K v ››› “ | v ^ v i j | ›› v i j ›› . Rearranging this we get | v ^ v i ^ ¨ ¨ ¨ ^ v i d ´ |} v i j } ď | v ^ v i j || v i ^ ¨ ¨ ¨ ^ v i d ´ | . (14)Applying this to the terms on the left hand side reduces things to proving that d ÿ j “ | v ^ v ^ ¨ ¨ ¨ ^ p v j ^ ¨ ¨ ¨ ^ v d |} v j } ď d ÿ j “ | v ^ v j || v ^ ¨ ¨ ¨ ^ p v j ^ ¨ ¨ ¨ ^ v d | . Without loss of generality we may assume that | v ^ v |} v } ď ¨ ¨ ¨ ď | v ^ v d |} v d } , then bearing this in mind we can apply (14) to each term on the left hand side to get d ÿ j “ | v ^ v ^ ¨ ¨ ¨ ^ p v j ^ ¨ ¨ ¨ ^ v d |} v j } ď | v ^ v |} v } ˜ d ÿ j “ | v ^ ¨ ¨ ¨ ^ p v j ^ ¨ ¨ ¨ ^ v d |} v j } ¸ ` | v ^ v |} v } | v ^ ¨ ¨ ¨ ^ v d |} v } . d ÿ j “ | v ^ ¨ ¨ ¨ ^ p v j ^ ¨ ¨ ¨ ^ v d |} v j } ě | v ^ ¨ ¨ ¨ ^ v d |} v } , then by a simple application of the rearrangement inequality for numbers and our assumption on thesizes of the terms | v ^ v j |} v j } , we see that | v ^ v |} v } ˜ d ÿ j “ | v ^ ¨ ¨ ¨ ^ p v j ^ ¨ ¨ ¨ ^ v d |} v j } ¸ ` | v ^ v |} v } | v ^ ¨ ¨ ¨ ^ v d |} v }ď | v ^ v |} v } | v ^ ¨ ¨ ¨ ^ v d |} v } ` | v ^ v |} v } ˜ d ÿ j “ | v ^ ¨ ¨ ¨ ^ p v j ^ ¨ ¨ ¨ ^ v d |} v j } ¸ ď d ÿ j “ | v ^ v j || v ^ ¨ ¨ ¨ ^ p v j ^ ¨ ¨ ¨ ^ v d | , which is what we want. It suffices to prove the following claim: Claim . For any u , . . . , u d P R d we have d ÿ j “ | u ^ ¨ ¨ ¨ ^ p u j ^ ¨ ¨ ¨ ^ u d |} u j } ě | u ^ ¨ ¨ ¨ ^ u d |} u } . Proof of Claim 1.
Supposing that all the vectors lie in a p d ´ q -dimensional subspace, then it is clearthat they must be linearly dependent. In particular, one can find coefficients α , . . . , α d P R , which arenot all equal to zero, such that d ÿ j “ α j u j “ . (15)By assumption, we have ř dj “ | α j | ‰ , so by setting β j “ | α j | ř dj “ | α j | , and ˜ u j “ sgn p α j q u j , we can rewrite (15) as d ÿ j “ β j ˜ u j “ . (16)By definition, we have d ÿ j “ β j “ , so applying Proposition 1 we can write β j “ | ˜ u ^ ¨ ¨ ¨ ^ ˜ u j ´ ^ ˜ u j ` ^ ¨ ¨ ¨ ^ ˜ u d | Vol p ∆ ˜ u ,..., ˜ u d q “ | u ^ ¨ ¨ ¨ ^ u j ´ ^ u j ` ^ ¨ ¨ ¨ ^ u d | Vol p ∆ ˜ u ,..., ˜ u d q , ∆ ˜ u ,..., ˜ u d denotes the simplex with vertices ˜ u , . . . , ˜ u d . So, (16) becomes d ÿ j “ | u ^ ¨ ¨ ¨ ^ u j ´ ^ u j ` ^ ¨ ¨ ¨ ^ u d | Vol p ∆ ˜ u ,..., ˜ u d q ˜ u j “ , and then multiplying through by Vol p ∆ ˜u ,..., ˜u d q we get d ÿ j “ | u ^ ¨ ¨ ¨ ^ u j ´ ^ u j ` ^ ¨ ¨ ¨ ^ u d | ˜ u j “ . By the reverse triangle inequality, we have “ ››››› d ÿ j “ | u ^ ¨ ¨ ¨ ^ u j ´ ^ u j ` ^ ¨ ¨ ¨ ^ u d | ˜ u j ››››› ě | u ^ ¨ ¨ ¨ ^ u d |} ˜ u } ´ ››››› d ÿ j “ | u ^ ¨ ¨ ¨ ^ u j ´ ^ u j ` ^ ¨ ¨ ¨ ^ u d | ˜ u j ››››› . Now by simply rearranging and applying the standard triangle inequality, we see that | u ^ ¨ ¨ ¨ ^ u d |} u } “ | u ^ ¨ ¨ ¨ ^ u d |} ˜ u } ď ››››› d ÿ j “ | u ^ ¨ ¨ ¨ ^ u j ´ ^ u j ` ^ ¨ ¨ ¨ ^ u d | ˜ u j ››››› ď d ÿ j “ | u ^ ¨ ¨ ¨ ^ u j ´ ^ u j ` ^ ¨ ¨ ¨ ^ u d |} ˜ u j }“ d ÿ i “ | u ^ ¨ ¨ ¨ ^ u j ´ ^ u j ` ^ ¨ ¨ ¨ ^ u d |} u j } . So it suffices to reduce to the case where all the vectors lie in a p d ´ q -dimensional subspace. Supposethat the vectors u , . . . , u d do not lie in a p d ´ q -dimensional subspace. Let us define a new family ω , . . . , ω d of vectors from the original family by simply projecting u onto the subspace spanned by u , . . . , u d and scaling appropriately. More precisely define ω : “ } u }} πu } u , where π “ π span t u ,...,u d u and for j “ , . . . , d ω j : “ u j . Clearly | ω ^ ¨ ¨ ¨ ^ ω d |} ω } “ | u ^ ¨ ¨ ¨ ^ u d |} u } , so it suffices to prove that for j “ , . . . , d | ω ^ ¨ ¨ ¨ ^ x ω j ^ ¨ ¨ ¨ ^ ω d |} ω j } ď | u ^ ¨ ¨ ¨ ^ p u j ^ ¨ ¨ ¨ ^ u d |} u j } , which follows from the fact that | πu ^ u ^ ¨ ¨ ¨ ^ p u j ^ ¨ ¨ ¨ ^ u d |} πu } ď | u ^ ¨ ¨ ¨ ^ p u j ^ ¨ ¨ ¨ ^ u d |} u } This concludes the proof of Lemma 5.By a simple argument we can now establish p q for k “ in all dimensions:12 roof of Lemma 4. Directly applying Lemma 5 with d “ , for any v , . . . , v P R we get | v ^ v ^ v |} v } ` | v ^ v ^ v |} v } ` | v ^ v ^ v |} v } ď | v ^ v || v ^ v |`| v ^ v || v ^ v |` | v ^ v || v ^ v | . For higher dimensions simply note that by the d “ case we have ÿ | v ^ v a ^ v b |} v c } “ ÿ t i,j,k uĂr d s | v ^ v j ^ v k |} v i } ` | v ^ v i ^ v k |} v j } ` | v ^ v i ^ v j |} v k }ď ÿ t i,j,k uĂr d s | v j ^ v k || v ^ v i | ` | v i ^ v k || v ^ v j | ` | v i ^ v j || v ^ v k |“ ÿ | v a ^ v b || v ^ v c | . As mentioned in the introduction, the vector-valued Maclaurin inequality with p “ can be rewrittenas a sequence of inequalities between intrinsic volumes of certain polytopes. Firstly, let us state a usefulformula for calculating the mixed volume of a zonoid: Theorem 8 (Theorem 5.3.2 in [16]) . For i “ , . . . , j ď d let Z j be a generalised zonoid with generatingmeasure ρ i and let K , . . . , K d ´ j Ă R d be convex bodies. Then V p Z , . . . , Z j , K , . . . , K d ´ j q “ j p d ´ j q ! d ! ż S d ´ ¨ ¨ ¨ ż S d ´ | u ^ ¨ ¨ ¨ ^ u j | v p d ´ j q ` π t u ,...,u j u K K , . . . , π t u ,...,u j u K K d ´ j ˘ dρ p u q ¨ ¨ ¨ dρ j p u j q , where v p d ´ j q denotes the j -dimensional mixed volume. Note that any zonotope Z “ m ř i “ α i r´ v i , v i s has a support function defined by h Z p u q “ m ÿ i “ α i |x u, v i y| “ ż S d ´ |x u, v y| dρ p v q , where ρ is concentrated at ˘ v i and assigns mass α i to each of these points. So by Theorem 8, givenzonoids Z “ ř k α k r´ v k , v k s , . . . , Z j “ ř k j α k j r´ v k j , v k j s and a fixed convex body K Ă R d , we have V p Z , . . . , Z j , K r d ´ j sq “ j p d ´ j q ! d ! ÿ k ,...,k j α k ¨ ¨ ¨ α k j ˇˇ v k ^ ¨ ¨ ¨ ^ v k j ˇˇ ˇˇˇ π Kt v k ,...,v kj u K ˇˇˇ In particular, if P is a centred parallelotope with edges of lengths } v } , . . . , } v d } in the directions of v } v } , . . . , v d } v d } , then we have h P p u q “ d ÿ i “ |x u, v i y| . It follows that V k p P q “ ˆ dk ˙ V p P, k ; B d , d ´ k q ω d ´ k “ k p d ´ k q ! d ! ř t i ,...,i k uĂr d s ` ˘ k | v i ^ ¨ ¨ ¨ ^ v i k | ˇˇˇ π t v i ,...,v ik u K B d ˇˇˇ ω d ´ k “ ˆ dk ˙ k ! p d ´ k q ! d ! ÿ ď i 㨨¨ă i k ď d | v i ^ ¨ ¨ ¨ ^ v i k |“ ÿ ď i 㨨¨ă i k ď d | v i ^ ¨ ¨ ¨ ^ v i k | .
13n this language, Conjecture 1 states that for all parallelotopes P Ă R d and ă j ď d , the followinginequality holds ˜ V j p P q ` dj ˘ ¸ j ď ˜ V j ´ p P q ` dj ´ ˘ ¸ j ´ , (17)with equality if and only if P is a cube. Suppose instead we consider the zonotope Z “ m ř i “ r´ v i , v i s Ă R d ,then by the same argument we have V k p Z q “ ˆ dk ˙ V p Z, k ; B d , d ´ k q ω d ´ k “ k p d ´ k q ! d ! ř t i ,...,i k uĂr m s ` ˘ k | v i ^ ¨ ¨ ¨ ^ v i k | ˇˇˇ π t v i ,...,v ik u K B d ˇˇˇ ω d ´ k “ ˆ dk ˙ k ! p d ´ k q ! d ! ÿ ď i 㨨¨ă i k ď m | v i ^ ¨ ¨ ¨ ^ v i k |“ ÿ ď i 㨨¨ă i k ď m | v i ^ ¨ ¨ ¨ ^ v i k | . So for k ď d we can also write Conjecture 1 as ˜ V j p Z q ` mj ˘ ¸ j ď ˜ V j ´ p Z q ` mj ´ ˘ ¸ j ´ . (18)Rearranging this gives V j ´ p Z q j V j p Z q j ´ ď ` mj ´ ˘ j ` mj ˘ j ´ . (19)This can be compared now to the Aleksandrov inequalitites for quermassintegrals: ˆ W i p K q| B n | ˙ n ´ i ě ˆ W j p K q| B n | ˙ n ´ j n ą i ą j ě . Taking into account that V j p K q “ ` nj ˘ W n ´ j p K q κ n ´ j we can rewrite the last one as V j ´ p K q j V j p K q j ´ ď V j ´ p B d q j V j p B d q j ´ . (20)Using the Aleksandrov inequality, McMullen proved the following dimension-free bound for the intrinsicvolumes. Namely, V j p K q ě j ` j V j ` p K q V j ´ p K q . Suppose now that Conjecture 1 is true. Then, we will show that we can get a stronger McMullen-typeinequality for zonotopes. We can use (19) to recover the above. Indeed, (19) implies log-concavity, V j p Z q ě ` mj ˘ ` mj ´ ˘` mj ` ˘ V j ` p Z q V j ´ p Z q . This can be simplified to V j p Z q ě p j ` qp m ´ j ` q j p m ´ j q V j ` p Z q V j ´ p Z q . The factor that appears in the right-hand side is always at least j ` j , which implies McMullen’s inequality.Moreover, we attain this bound as m Ñ 8 , as expected.14ote also that in [9] the following inequalities were proved: for zonoids Z sup Λ P GL p d q V j p Λ Z q V p Λ Z q j ě d j ˆ dj ˙ (21)if dim p Z q “ d and j ě , with equality if and only if Z is a parallelotope. These are reverse, in a sense,to a consequence of (20), namely for any convex body KV j p K q V p K q j ď V j p B d q V p B d q j . (22)The equality case in (21) exactly says that ˜ V j p P q ` dj ˘ ¸ j ď V p P q ` d ˘ . (23)This is of course a special case of (17). If this more general equality case can be established, one mighthope to prove the corresponding generalisation to (21). p “ Let us see how we can use this language borrowed from convex geometry to reinterpret the reductiondescribed at the beginning of Section 4. In particular, let us try to rewrite (12) purely in terms ofintrinsic volumes. Firstly, fix v , . . . , v m ´ P R m ´ Ă R m and u P R m ´ ˆ R – R m , then setting Z “ ř m ´ i “ r´ v i , v i s Ă R m ´ , we can apply Theorem 8 to get V k p π u K Z q “ ˆ m ´ k ˙ V ` π u K Z, k ; B m ´ , m ´ k ´ ˘ ω m ´ k ´ “ ˆ mm ´ ˙ˆ m ´ k ˙ V ` Z, k ; B m , m ´ k ´ r´ u, u s ˘ ω m ´ k ´ “ p m q ! p m ´ k ´ q ! k ! ´ k ` p m ´ k ´ q ! m ! ¯ ř t i ,...,i k uĂr m ´ s ` ˘ k ` | v i ^ ¨ ¨ ¨ ^ v i k ^ u | ˇˇˇ π t v i ,...,v ik ,u u K B m ˇˇˇ ω m ´ k ´ “ ÿ ď i 㨨¨ă i k ď m ´ | v i ^ ¨ ¨ ¨ ^ v i k ^ u | . For any u , . . . , u m P R m , if we set Z “ ř mi “ r´ u i , u i s , then using the calculation we have just madealong with Theorem 8, inequality (12) can be rewritten as V k ´ ´ π u K Z ¯ V k ´ p Z q ď V k ´ ´ π u K Z ¯ V k ´ p Z q (24)for ă k ď m ´ . Theorem 1.2 in [5] implies the following result: let K Ă R d be a convex body, thenfor any u P R d we have V k ´ p π u K K q V k ´ p K q ď p d ´ k ` q d ´ k ` V k ´ p π u K K q V k ´ p K q , (25)for all ď k ď d . In the special case where m “ d ` and u lies in the span of u , . . . , u m , one can view(24) as the sharp version of (25) when we restrict ourselves to zonotopes. Using an analogous derivationto the one given at the beginning of Section 4, we can deduce the following result as a consequence of(25): Theorem 9.
For any d -tuple of vectors v , . . . , v d P R d and any ă k ď d , the following inequalityholds: ¨˚˝ ř ď i 㨨¨ă i k ď d | v i ^ ¨ ¨ ¨ ^ v i k | ` dk ˘ ˛‹‚ k ď p d ´ k ` qp d ´ k ` q ¨˚˝ ř ď i 㨨¨ă i k ´ ď d | v i ^ ¨ ¨ ¨ ^ v i k ´ | ` dk ´ ˘ ˛‹‚ k ´ , but smaller than . As aconsequence of Theorem 5, we know that the sharp constant is equal to in dimensions d “ , . Thus,it seems likely that the constant given in Theorem 9 is suboptimal. Acknowledgements.
The authors would like to thank Anthony Carbery and Apostolos Giannopou-los for useful discussions. The first named is supported by the Hellenic Foundation for Research andInnovation (Project Number: 1849).
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Keywords:
Maclaurin inequality, andov-Fenchel inequality, mixed volumes, parallelotopes.
Primary 52A20; Secondary 52A39, 15A45.
Silouanos Brazitikos : Department of Mathematics, National and Kapodistrian University of Athens, Panepistimiopolis157-84, Athens, Greece.
E-mail: [email protected]
Finlay McIntyre : School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh,JCMB, Peter Guthrie Tait Road King’s Buildings, Mayfield Road, Edinburgh, EH9 3FD, Scotland.
E-mail: [email protected]@sms.ed.ac.uk