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The theorems of Caratheodory and Gluskin for 0 < p < 1

arXiv:math/9209213v1 [math.FA] 10 Sep 1992Preliminary version.The theorems of Caratheodory and Gluskin for 0 < p < 1byJes´us Bastero * , Julio Bernu´es * and Ana Pe˜na **Departamento de Matem´aticas. Facultad de CienciasUniversidad de Zaragoza50009-Zaragoza (Spain)1.

Introduction and notation.Throughout the paper X will denote a real vector space and p will be a realnumber, 0 < p < 1. A set A ⊆X is called p-convex if λx + µy ∈A, wheneverx, y ∈A, and λ, µ ≥0, with λp + µp = 1.

Given A ⊆X, the p-convex hull of A isdefined as the intersection of all p-convex sets that contain A. Such set is denotedby p-conv (A).A p-norm on X is a map ∥· ∥: X →IR verifying:(i) ∥x∥≥0, ∀x ∈X and ∥x∥= 0 ⇔x = 0.

(ii) ∥ax∥= |a| ∥x∥, ∀a ∈IR, x ∈X. (iii) ∥x + y∥p ≤∥x∥p + ∥y∥p, ∀x, y ∈X.We will say that (X, ∥·∥) is a p-normed space.

The unit ball of a p-normed spaceis a p-convex set and will be denoted by BX.We denote by Mpn the class of all n-dimensional p-normed spaces. If X, Y ∈Mpnthe Banach-Mazur distance d(X, Y ) is the infimun of the products ∥T∥·∥T −1∥, wherethe infimun is taken over all the isomorphisms T from X onto Y .We shall use the notation and terminology commonly used in Banach spacetheory as it appears in [Tmcz].In this note we investigate some aspects of the local structure of finite dimen-sional p-Banach spaces.The well known theorem of Gluskin gives a sharp lowerbound of the diameter of the Minkowski compactum.In [Gl] it is proved thatdiam(M1n) ≥cn for some absolute constant c. Our purpose is to study this problemin the p-convex setting.

In [Pe], T. Peck gave an upper bound of the diameter of Mpnnamely, diam(Mpn) ≤n2/p−1. We will show that such bound is optimum.The method used by Gluskin to prove his result can be directed applied, withsome minor variations, to our case.

At some point of the proof it is necessary to findsome volumetric estimates for convex envelopes. In particular if {Pi}mi=1 are m pointsin the euclidean sphere in IRn we need to estimate from above|conv {±Pi}||Bℓn2 |1/n.

* Partially supported by Grant DGICYT PS 90-0120** Supported by Grant DGA (Spain)1

Szarek, [Sz], and other authors gave the estimate|conv {±Pi}||Bℓn2 |1/n≤Cmn−3/2(C is an absolute constant). Caratheodory’s convexity theorem turned out to be animportant ingredient of the proof.

For p < 1 we will proceed in this fashion and sowe will need to have the corresponding version of Caratheodory convexity theorem.The main results of the paper are the following:Theorem 1. Let A ⊆IRn and 0 < p < 1.For every x ∈p-conv (A), x ̸= 0there exist linearly independent vectors {P1 .

. .

Pk} ⊆A with k ≤n, such thatx ∈p-conv {P1 . .

.Pk}. Moreover, if 0 ∈p-conv (A), there exits {P1 .

. .

Pk} ⊆A withk ≤n + 1 such that 0 ∈p-conv {P1 . .

. Pk}.Theorem 2.

Let 0 < p < 1. There exits a constant Cp > 0 such that for everyn ∈INCpn2/p−1 ≤diam(Mpn) ≤n2/p−1.The first result can be viewed as the p-convex analogue of Caratheodory’s theo-rem.

Apparently, the result for p < 1 is stronger than the Caratheodory’s one in thesense that we get k ≤n and only k ≤n+1 can be assured for p = 1 (see [Eg], pg 35).Proposition 3 ii) will show that this is not such since vector 0 plays a particularlyspecial role.The second result is the analogue of Gluskin’s theorem in the p-convex setting,that is the diameter of the Minkowski compactum grows asimpthotically like n2/p−1.2. Caratheodory’s theorem for p-convex hulls.In this section we want to prove Theorem 1.

We begin by recalling the firstproperties of p-convex hulls. They are probably known but since we have not foundthem in any reference we sketch their proofs.Proposition 3.

For every ∅̸= A ⊆X.i)p-conv (A) =( nXi=1λixi | λi ≥0,nXi=1λpi = 1, xi ∈A, n ∈IN).ii)p-conv (A ∪{0}) = {0}[p-conv (A)=( nXi=1λixi | λi ≥0,nXi=1λpi ≤1, xi ∈A, n ∈IN)iii)p-conv (T(A)) = T(p-conv (A)), for any linear map T.Proof: i) and iii) are straighforward.ii) We only have to prove that p-conv (A∪{0}) ⊆{0}∪p-conv (A). It is enoughto show that every non zero element x of the form x =nXi=1λixi, xi ∈A,nXi=1λpi < 1can be written as x =mXi=1µiyi, yi ∈A,mX1µpi = 1.2

Suppose λ1 ̸= 0. Write λ1 =kXi=1βi, with βi ≥0.

We havenXi=1λpi ≤kXi=1βpi +nXi=2λpi ≤k1−pλp1 +nXi=2λpi .It is now clear, by a continuity argument, that we can find k and βi ≥0, 1 ≤i ≤k,such that λ1 =kXi=1βi andkXi=1βpi +nXi=2λpi = 1.Finally, the representation x =kXi=1βixi +nXi=2λixi does the job.///Remark. In particular ii) says that for every 0 ̸= x ∈X, p-conv {x} = (0, x] ={λx; 0 < λ ≤1}.

This situation is rather different from the case when p = 1.Another useful particular case of ii) is the following: If A = {P1, . .

., Pn} ⊂X, Pi ̸= 0, Pi ̸= Pj, ∀1 ≤i ̸= j ≤n, then 0 ̸= x ∈p-conv (A) ⇒x =nXi=1λiPi, withPni=1 λpi ≤1, λi ≥0. Observe that we allow no more than n non-zero summandswhile in i) and ii) there is no restriction.Next, we are going to prove a particular case of Theorem 1, which will help usin the general case.Lemma 4.

Let {P1 . .

. Pn, Q} ⊆IRn with {Pi}ni=1 linealy independent.If M ∈p-conv {P1 .

. .

Pn, Q, 0} then there exist Pi1 . .

. Pin ∈{P1 .

. .

Pn, Q} such that M ∈p-conv {Pi1 . .

. Pin, 0}.Proof: By Proposition 3 iii), it’s enough to consider the case {e1 .

. .

en, Q, 0}where {ei}ni=1 is the canonical basis in IRn and Q = (a1 . .

. an) ̸= 0.

Denote by P thesubset of p-conv {e1 . .

. en, Q, 0} for which the thesis of the Lemma holds.LetK = {(λ1 .

. .λn) ∈IRn |nXi=1λpi ≤1, λi ≥0, 1 ≤i ≤n}Write µ = µ(λ1 .

. .

λn) = (1 −nXi=1λpi )1/p, and consider the map ϕ: K →IRn definedas λ = (λ1 . .

.λn) →ϕ(λ) =nXi=1λiei + µQ. Denote by Jϕ(λ) the Jacobian of thefunction ϕ at a point λ.The proof of the Lemma rests on the following:Claim.

For every λ ∈Int(K) such that Jϕ(λ) = 0 we have ϕ(λ) ∈P.Assume the claim is true and continue with the proof of the Lemma.3

Let M ∈p-conv {e1 . .

. en, Q, 0} i.e.M = Pni=1 λiei + νQ, with λi, ν ≥0,1 ≤i ≤n, Pni=1 λpi + νp ≤1.Suppose first that Pni=1 λpi + νp = 1, that is M = ϕ(λ), λ ∈K.

If λ ∈∂(K)then, either λi or ν are equal to 0 and clearly ϕ(λ) ∈P. If λ ∈Int(K), we also havetwo posibilities: a) Jϕ(λ) = 0 and the claim says that M ∈P or b) Jϕ(λ) ̸= 0.

Bythe inverse function theorem we necessarily have that M ∈Intϕ(K). Since ϕ(K) iscompact there exists t > 1 such that tM ∈∂ϕ(K), and therefore tM = ϕ(λ′) witheither λ′ ∈∂(K) or λ′ ∈Int(K) and Jϕ(λ′) = 0.

In any case we deduce that tMbelongs to P and so does M.If, on other hand,nXi=1λpi + νp = sp < 1 the results easily follows by consideringMs and applying the preceding case.///Proof of the Claim: It is an easy exercise to show that the Jacobian of ϕ isJϕ(λ) = 1 −nXi=1ai µλi1−p.For every λ ∈Int(K) we write λ = tv, v = (v1 . .

. vn), vi > 0, 1 ≤i ≤n,0 < t < 1, Pni=1 vpi = 1.

We have Jϕ(λ) = 0 if and only if µλi1−pnXi=1aiv1−pi= 1andµp = 1 −tpWrite R = nXi=1aiv1−pi!11−p> 0. It is easy to see that for every v, there is a uniquet ∈(0, 1) such that Jϕ(tv) = Jϕ(λ) = 0.

Explicitly λ =R(1 + Rp)1/p v. Therefore withthis new notation, the points λ with Jϕ(λ) = 0 are such that M =nXi=1Rvi + ai(1 + Rp)1/p eiwhere vi > 0, 1 ≤i ≤n,nXi=1vpi = 1, R1−p =nXi=1aiv1−pi> 0.Case 1. If Rvi + ai ≥0 for all i, then M ∈p-conv {e1 .

. .

en, 0}. Indeed, let’s showthatnXi=1(Rvi + ai)p < 1 + RpThis is equivalent tonXi=1vi + aiRp−nXi=1vpi −1RnXi=1aiv1−pi< 04

and tonXi=1vpi1 + aiRvip−nXi=1vpi1 + aiRvi< 0.But this is obvious by the elementary inequality:(1 + x)p ≤1 + px,x ≥−1(∗)Case 2. If there is some i, 1 ≤i ≤n such that Rvi + ai < 0, then ai < 0.

Wesuppose without loss of generality that min{aivi| 1 ≤i ≤n} is achieved at i = 1.We shall prove M ∈p-conv {e2 . .

.en, Q, 0}. Recall that M =nXi=1Rvi + ai(1 + Rp)1/p ei.

Wewill show that M can be represented as M =nXi=2αiei + βQ withnXi=2αpi + βp < 1.Comparing the two representations, it is easy to see thatαi =vi −v1aia1R(1 + Rp)1/p , 2 ≤i ≤nβ =1 + Rv1a11(1 + Rp)1/p .By hypothesis we have β, αi ≥0, 2 ≤i ≤n. It remains to show thatnXi=2αpi +βp < 1, which is the same asnXi=2vi −v1aia1p+1 + Rv1a1p 1Rp < 1 + 1RnXi=1aiv1−pior nXi=1aiRv1−pi!

1 + Rv1a1p+nXi=2vpi1 −v1aia1vip−nXi=1vpi1 + aiRvi< 0Again (*) establishes1 + Rv1a1p≤1 + pRv1a1and1 −v1aia1vip≤1 −pv1aia1viand the result easily follows.///We are now ready to state and prove the main theorem of the section.Theorem 1. Let A ⊆IRn and 0 < p < 1.For every x ∈p-conv (A), x ̸= 0there exist linearly independent vectors {P1 .

. .

Pk} ⊆A with k ≤n, such thatx ∈p-conv {P1 . .

.Pk}. Moreover, if 0 ∈p-conv (A), there exits {P1 .

. .

Pk} ⊆A withk ≤n + 1 such that 0 ∈p-conv {P1 . .

. Pk}.Proof: Let x ∈p-conv (A), x ̸= 0, then x = PNi=1 λiPi with Pi ∈A, Pi ̸= 0,PNi=1 λpi ≤1, λi > 0 and 1 ≤i ≤N.Let dim(span{Pi}Ni=1) = m ≤n.By5

Proposition 3 iii) and without loss of generality, we can suppose that we are in Rmand that x =NXi=1λiPi with P1 . .

. Pm linearly independent.Write sp =m+1Xi=1λpi and ˜x =m+1Xi=1λis Pi.

Clearly ˜x ∈p-conv {P1 . .

. Pm+1} andtherefore, by Lemma 4 there exists {Pk1 .

. .

Pkm} ⊂{P1 . .

. Pm+1} such that ˜x =mXi=1βiPki,mXi=1βpi ≤1.

Hencex = s˜x +NXi=m+2λiPi =mXi=1sβiPki +NXi=m+2λiPiwithmXi=1βpi sp +NXi=m+2λpi ≤sp +NXi=m+2λpi ≤1.We have represented x as a combination of points of A of length N −1. Con-sider now, span{Pk1 .

. .

Pkm, Pm+2 . .

. PN} and repeat the argument until reaching arepresentation of length ≤n.If 0 ∈p-conv (A) then 0 =NXi=1λiPi, Pi ∈A, λi > 0, 1 ≤i ≤N andNXi=1λpi = 1.As before, we can suppose P1 .

. .

Pm linearly independent with m ≤n. We considerm+1Xi=1λiPi = −NXi=m+2λiPi.

If we apply Lemma 4 to ˜x =m+1Xi=1λis Pi, sp =m+1Xi=1λpi weobtainmXi=1βiPi = −NXi=m+2λiPiwith Pmi=1 βpi ≤1.Hence 0 ∈p-convex envelope of N −1 points.Repeat theargument until reaching a representation of length ≤n + 1.///3. Gluskin’s theorem for 0 < p < 1.In this section we are going to prove Theorem 2.

As quoted above, Peck showedthat diam (Mpn) ≤n2/p−1. Given an n-dimensional p-normed space X, he consideredits Banach envelope Xb (the normed space whose unit ball is the convex envelopeof the unit ball of X) and proved d(X, Xb) ≤n1/p−1 (see [Pe] or [G-K]).

By usingJohn’s theorem he obtained the estimate. We want to prove that this result is sharp.More precisely what we are going to show isTheorem 2.

Let 0 < p < 1. There exits a constant Cp > 0 such that for everyn ∈INCpn2/p−1 ≤diam(Mpn) ≤n2/p−1.6

The proof of Theorem 2 follows Gluskin’s original ideas.We first introducesome notation. Sn−1 will denote the euclidean sphere in IRn with its normalizedHaar measure µn−1 and Ωwill be the product space Sn−1×n).

. .

×Sn−1 endowedwith the product probability IP. If K ⊆IRn, |K| is the Lebesgue measure of K.If A = (P1, .

. ., Pn) ⊂Ω, we write Qp(A) = p-conv {±ei, ±Pi | 1 ≤i ≤n}, being{ei}ni=1 the canonical basis of IRn.

We denote by ∥· ∥Qp(A) the p-norm in IRn whoseunit ball is Qp(A).We only need to prove that for some absolute constant Cp > 0, there existA, A′ ∈Ωsuch that simultaneously∥T∥Qp(A)→Qp(A′) ≥Cpn1/p−1/2and∥T −1∥Qp(A′)→Qp(A) ≥Cpn1/p−1/2hold for any T ∈SL(n) (that is, any linear isomorphism in IRn with det T = 1).Straightforward argument shows that it is enough to see that for any fixed A′ ∈Ωwe have,IP{ A ∈Ω| ∥T∥Qp(A)→Qp(A′) < Cpn1/p−1/2 for some T ∈SL(n) } < 12Fix A ∈Ωand t > 0. Consider the setΩ(A′, t) = { A ∈Ω| ∥T∥Qp(A)→Qp(A′) < t for some T ∈SL(n) }The proof of the following three lemmas are analogous to the ones in the casep = 1 (see [Tmcz], §38).Lemma 5.

Let A′ ∈Ωand t > 0. There exists a t-net N(A′, t) in { T ∈SL(n) |∥T∥ℓnp →Qp(A′) ≤t} with respect to the metric induced by ∥·∥pℓn2 →Qp(A′) of cardinality| N(A′, t) | ≤(3n1/p−1/2)n2|Qp(A′)|n| {T ∈SL(n) | ∥T∥ℓn2 →ℓn2 ≤1} |Lemma 6.

For every A′ ∈Ωand t > 0 we have,Ω(A′, t) ⊆[T ∈N(A′,t){ A ∈Ω| ∥T(Pi)∥Qp(A′) ≤21/pt, ∀1 ≤i ≤n }Lemma 7. Given T ∈SL(n), A′ ∈Ωand t > 0,IP{ A ∈Ω| ∥T(Pi)∥Qp(A′) ≤21/pt, ∀1 ≤i ≤n } ≤(21/pt)n2 | Qp(A′) ||Bℓn2 |nProof of Theorem 2: Numerical constants are always denoted by the sameletters C (or Cp, if it depends only on p) although they may have different value7

from line to line. Using consecutively the three preceding lemmas we have for everyA′ ∈Ωand t > 0,IPΩ(A′, t)≤(Cptn1/p−1/2)n2|Qp(A′)|2n|Bℓn2 |n · | {T ∈SL(n) | ∥T∥ℓn2 →ℓn2 ≤1} |It is well known that for some absolute constant C > 0, (see [Tmcz]),| {T ∈SL(n) | ∥T∥ℓn2 →ℓn2 ≤1} | ≥Cn2|Bℓn2 |nNow using Theorem 1 it is clear that if A′ = {P1, .

. .Pn}, then Qp(A′) ⊆[p-conv {Pk1, .

. ., Pkn} where the union runs over the4nnchoices of {Pki}ni=1 ⊆{±ei, ±Pi, 1 ≤i ≤n}.

Since ∥Pi∥2 = 1 and|p-conv {Pk1, . .

., Pkn}| = |det [Pk1, . .

., Pkn] | · |p-conv {e1, . .

., en}|we get|Qp(A′)| ≤4nn|Bℓnp |2n≤Cnp n−n/p2−nfor some constant Cp (see [Pi], pg 11). Hence,IPΩ(A′, t)≤(Cptn1/2−1/p)n2If we take a suitable t > 0, we can assure IPΩ(A′, t)< 12 and the result follows.///Remark.

In the same way as quoted above, given a p-normed space X and p

Indeed, for every x ∈BXq = q-conv (BX) and ∥x∥Xq = 1 thereexist P1, . .

., Pn ∈BX such that x = Pni=1 λiPi with λi ≥0, 1 ≤i ≤n, Pni=1 λqi ≤1and1 ≤∥x∥X ≤nXi=1λpi ∥Pi∥pX ≤nXi=1λpi ≤n1/p−1/qby homogeneity we achieve the result. Now it is easy to see that if X, Y are the spacesappearing in Theorem 2, then d(X, Xq) ≥Cpn1/p−1/q, d(Y, Y q) ≥Cpn1/p−1/q andd(Xq, Y q) ≥Cpn2/q−1.

In particular, for q = 1, d(X, Xb) ≥Cpn1/p−1 , d(Y, Y b) ≥Cpn1/p−1 and d(Xb, Y b) ≥Cpn.Acknowledgments. The authors are indebted to Yves Raynaud for some commentsin the proof of Lemma 4.8

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: Convexity. Cambridge Tracts in Math.

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and Appl. 15(1), 72-73 (1981).

[G-K] Gordon, Y., Kalton, N.J.: Local structure for quasi-normed spaces. Preprint,(1992).

[Pe] Peck, T.: Banach-Mazur distances and projections on p-convex spaces. Math.Zeits.

177, 132-141 (1981). [Pi] Pisier G.: The volume of convex bodies and Banach Space Geometry.

CambridgeUniversity Press (1989). [Sz] Szarek, S.J.

: Volume estimates and nearly Euclidean decompositions of normedspaces. S´eminaire d’Analyse Fonctionnelle ´Ecole Poly.

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(1979-80)[Tmzc] Tomczak-Jaegerman, N.: Banach-Mazur distances and finite-dimensional oper-ator ideals. Pitman Monographs 38 (1989).9

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