CComputation of maximal projectionconstants
Giuliano BassoJanuary 24, 2019
Abstract
The linear projection constant Π ( E ) of a finite-dimensional real Ba-nach space E is the smallest number C ∈ [ + ∞ ) such that E is a C -absolute retract in the category of real Banach spaces with boundedlinear maps. We denote by Π n the maximal linear projection constantamongst n -dimensional Banach spaces. In this article, we prove that Π n may be determined by computing eigenvalues of certain two-graphs.From this result we obtain that the relative projection constants of codi-mension n converge to + Π n . Furthermore, using the classification of K -free two-graphs, we give an alternative proof of Π = . We alsoshow by means of elementary functional analysis that for each integer n (cid:62) there exists a polyhedral n -dimensional Banach space F n suchthat Π ( F n ) = Π n . As a consequence of ideas developed by Lindenstrauss, cf.[Lin64], for a finite-dimensional Banach space E ⊂ (cid:96) ∞ ( N ) the smallest con-stant C ∈ [ + ∞ ) such that E is an absolute C -Lipschitz retract is completelydetermined by the linear theory of E . Indeed, Rieffel, cf. [Rie06], establishedthat it is equal to the linear projection constant of E , which is the number Π ( E ) ∈ [ + ∞ ] defined asinf (cid:8) (cid:107) P (cid:107) | P : (cid:96) ∞ ( N ) → E bounded surjective linear map with P = P (cid:9) . Linear projections have been the object of study of many researchersand the literature can be traced back to the classical book by Banach, cf.[Ban32, p.244-245]. The question about the maximal value Π n of the linearprojection constants of n -dimensional Banach spaces has persisted and is anotoriously difficult one. In this article, we establish a formula that relates Π n with eigenvalues of certain two-graphs. This reduces the problem (inprinciple) to the classification of certain two-graphs and thus allows the1 a r X i v : . [ m a t h . M G ] J a n ntroduction of tools from graph theory. Following this approach, we presentan alternative proof of Π = , see 4.3, and we establish that that the relativeprojection constants of codimension n converge to + Π n , see Corollary 1.3.In the remainder of this overview, we summarize the current state of thetheory.For n (cid:62) , define Ban n to be the set of linear isometry classes of n -dimensional Banach spaces over the real numbers. The set Ban n equippedwith the Banach-Mazur distance is a compact metric space, cf. [TJ89]. Thus,the map log ◦ Π : Ban n → [ + ∞ ) is 1-Lipschitz and consequently for all n (cid:62) the maximal projection constant of order n , Π n := max (cid:8) Π ( X ) : X ∈ Ban n (cid:9) , is a well-defined real number. Apart form Π = , the only known valueis Π = , due to Chalmers and Lewicki, cf. [CL10]. There is numericalevidence indicating that Π = ( + √ ) /2 , cf. [FS17, Appendix B], but to theauthor’s knowledge, there is no known candidate for Π n for all n (cid:62) . Froma result of Kadets and Snobar, cf. [KS71], Π n (cid:54) √ n. Moreover, König, cf. [Kön85], has shown that this estimate is asymptoti-cally the best possible. Indeed, there exists a sequence ( X n k ) k (cid:62) of finite-dimensional real Banach spaces such that dim ( X n k ) = n k , where n k → + ∞ for k → + ∞ , and lim k → + ∞ Π ( X n k ) √ n k = There are many non-isometric maximizers of the function Π n ( · ) , cf. [KTJ03].A finite-dimensional Banach space is called polyhedral if its unit ball is a poly-tope. Equivalently, a finite-dimensional Banach space ( E, (cid:107)·(cid:107) ) is polyhedralif there exists an integer d (cid:62) such that ( E, (cid:107)·(cid:107) ) admits a linear isomet-ric embedding into (cid:96) d ∞ . Using a result of Klee, cf. [Kle60, Proposition 4.7],and elementary functional analysis, we show that there exist maximizers of Π n ( · ) that are polyhedral, see Theorem 1.4.In the 1960s, Grünbaum, cf. [Grü60], calculated Π ( (cid:96) n1 ) , Π ( (cid:96) n2 ) and Π ( X hex ) ,where X hex is the 2-plane with the hexagonal norm. In particular, Π ( X hex ) = , which Grünbaum conjectured to be the maximal value of Π ( · ) amongst -dimensional Banach spaces. In 2010, Chalmers and Lewicki presented anintricate proof of Grünbaum’s conjecture employing the implicit functiontheorem and Lagrange multipliers, cf. [CL10].Our main result, see Theorem 1.2, provides a characterization of thenumber Π n in terms of certain maximal sums of eigenvalues of two-graphsthat are K n + -free. In [FF84], Frankl and Füredi give a full description oftwo-graphs that are K -free. Via this description and Theorem 1.2 we canderive from first principles that Π = . This is done in Section 4.2ext, we introduce the necessary notions from the theory of two-graphsthat are needed to properly state our main result. The subsequent definition of a two-graph via cohomol-ogy follows Taylor [Tay77], and Higman [Hig73]; see also [Sei91, Remark4.10]. Let V denote a finite set. For each integer n (cid:62) we set E n ( V ) := (cid:8) B ⊂ V : | B | = n (cid:9) and E n ( V ) := (cid:8) f : E n ( V ) → F (cid:9) , where F denotes the field with two elements. Elements of E ( V ) are finitesimple graphs. If n is strictly greater than the cardinality of V , then E n ( V ) consists only of the empty function ∅ → F . For each f ∈ E n ( V ) the map δf ∈ E n + ( V ) is given by B (cid:55)→ (cid:88) v ∈ B f ( B \ { v } ) . Clearly, it holds that δ ◦ δ = , where denotes the neutral element of thegroup E n + ( V ) . Two-graphs can be defined as follows. Definition 1.1 (two-graph)
A two-graph is a tuple T = ( V, ∆ ) , where V and ∆ are finite sets and there exists a map f T ∈ E ( V ) such that δf T = and ∆ = f − ( ) . The cardinality of V is called the order of T . Among other things, two-graphs naturally occur in the study of systems ofequiangular lines and 2-transitive permutation groups; authoritative surveysare [Sei91; Sei92]. Given a two-graph T = ( V, ∆ ) , the following set is alwaysnon-empty: [ T ] := (cid:8) f : E ( V ) → F : δf = f T (cid:9) . Each f ∈ [ T ] gives rise to a graph G f := ( V, f − ( )) . The Seidel adjacencymatrix of a graph G = ( V, E ) is the matrix S ( G ) , which is the symmetric | V | × | V | -matrix given by S ( G ) ij = if i = j − if i and j are adjacent otherwise.For each choice f , f ∈ [ T ] the matrices S ( G f ) and S ( G f ) have the samespectrum.By definition, the eigenvalues of T = ( V, ∆ ) are the real numbers λ ( T ) (cid:62) . . . (cid:62) λ | V | ( T ) that are the eigenvalues of S ( G f ) for f ∈ [ T ] (counted with multiplicity). Thisdefinition is independent of f ∈ [ T ] .We say that a two-graph T = ( V, ∆ ) is K n -free if there is no injective map ϕ : {
1, . . . , n } → V such that (cid:8) ϕ ( v ) , ϕ ( v ) , ϕ ( v ) (cid:9) ∈ ∆ for all distinct points v , v , v ∈ V n . 3 .3 Main result Our main result reads as follows:
Theorem 1.2 If n (cid:62) is an integer, then Π n = sup d (cid:62) max (cid:14) nd + n (cid:88) k = λ k ( T ) : T is a K n + -free two-graph of order d (cid:15) . To prove Theorem 1.2, we invoke a simple trick, see Lemma 2.2, thatallows us to greatly narrow down the matrices that need to be considered.This is done in Section 2.
The following question has first beensystematically addressed by König, Lewis, and Lin in [KLL83]:
Question 1.1
Let n, d (cid:62) be integers. What is Π ( n, d ) := sup (cid:8) Π ( E ) : E ⊂ (cid:96) ∞ d is an n -dimensional Banach space (cid:9) ?By definition, sup ∅ = − ∞ . Clearly, Π ( d, d ) = and it is a direct conse-quence of the classical Hahn-Banach theorem that Π (
1, d ) = for all inte-gers d (cid:62) . The quantity Π ( d −
1, d ) has been examined by Bohnenblust, cf.[Boh38], where it is shown that Π ( d −
1, d ) (cid:54) − . In [CL09], Chalmers andLewicki determined the exact value of Π (
3, 5 ) . In [KLL83], König, Lewis, andLin established the general upper bound Π ( n, d ) (cid:54) nd + (cid:114)(cid:0) d − (cid:1) nd (cid:0) − nd (cid:1) with equality if and only if R n admits a system of d distinct equiangularlines. Thereby, as R admits a system of six equiangular lines, cf. [LS73, p.496], it holds that Π (
3, 6 ) = + √
52 .
In light of Π (
4, 6 ) =
53 , which we demonstrate in Paragraph 4.4, up to d = all exact values of Π ( n, d ) for (cid:54) n (cid:54) d are now computed. It is well-known that Π ( n, d ) (cid:54) Π ( n, d + ) and Π ( n, d ) (cid:54) Π ( n +
1, d + ) for all (cid:54) n (cid:54) d , cf. [CL09]. Via Theorem 1.2, we infer the following asymp-totic relation between these two increasing sequences: Corollary 1.3
For each integer n (cid:62) we have + Π n = lim d → + ∞ Π ( d − n, d ) . n = , then Corollary 1.3follows directly from the fact that Bohnenblust’s upper bound of Π ( d −
1, d ) is sharp, cf. [CL09, Lemma 2.6]. Recently, the special case n = has beenconsidered by Sokołowski in [Sok17].Recall that Π (
1, d ) = Π (
1, 1 ) = for all d (cid:62) . The proof of Grünbaum’sconjecture, cf. [CL10], shows that Π (
2, d ) = Π (
2, 3 ) = for all d (cid:62) Numerical experiments, cf. [FS17, Appendix B], suggest that if d ∈ {
6, . . . , 10 } ,then Π (
3, d ) = Π (
3, 6 ) . Since Π n ( · ) admits a polyhedral maximizer, the se-quence Π ( n, · ) stabilizes eventually. Theorem 1.4
Let n (cid:62) be an integer. There exists a polyhedral n -dimensionalBanach space ( F n , (cid:107)·(cid:107) ) such that Π ( F n ) = Π n . As a result, there is an integer D (cid:62) such that Π ( n, d ) = Π ( n, D ) for all d (cid:62) D . A proof of Theorem 1.4 can be found in Section 3. Π n Let d (cid:62) be an integer and set A d := (cid:8) d + S : S is a Seidel adjacency matrix of a simple graph of order d (cid:9) . Moreover, we use D d to denote the set of all diagonal d × d -matrices thathave trace equal to one and whose diagonal entries are non-negative.For A ∈ A d and D ∈ D d we write λ ( √ DA √ D ) (cid:62) . . . (cid:62) λ d ( √ DA √ D ) for the eigenvalues of the symmetric matrix √ DA √ D (counted with multi-plicity). The subsequent result, due to Chalmers and Lewicki, characterizesthe values Π ( n, d ) in terms of maximal sums of eigenvalues of matrices ofthe form √ DA √ D . Theorem 2.1 (Theorem 2.3 in [CL10])
Let (cid:54) n (cid:54) d be integers. The value Π ( n, d ) is attained and equals max (cid:14) n (cid:88) k = λ k (cid:16) √ DA √ D (cid:17) : A ∈ A d and D ∈ D d (cid:15) . .2 Blow-up of matrices Let i (cid:62) be an integer and consider the mapbl i : (cid:91) d (cid:62) i A d → (cid:91) d (cid:62) i + A d , A (cid:55)→ bl i ( A ) := (cid:20) A a ti a i (cid:21) where a i denotes the i -th row of A . By construction, the i -th row of bl i ( A ) and the last row of bl i ( A ) coincide. We say that the matrix bl i ( A ) is a blow-up of A (with respect to the i -th row).If A ∈ A d is a matrix and D ∈ D d is positive-definite, then all eigenval-ues of AD are real, for AD is equivalent to the symmetric matrix √ DA √ D .With a similar argument, one can show that even if D is positive-semidefinite,then all eigenvalues of AD are real. We use the notation λ ( AD ) := ( λ ( AD ) , . . . , λ d ( AD )) , where λ ( AD ) (cid:62) . . . (cid:62) λ d ( AD ) are the eigenvalues of AD (counted withmultiplicity). The lemma below is the key step in the proof of Theorem 1.2. Lemma 2.2
Let A (cid:48) ∈ A d − be a matrix, let A := bl i ( A (cid:48) ) for some integer (cid:54) i (cid:54) d − and let D := diag ( d , . . . , d d ) ∈ D d be an invertible matrix. Weset D (cid:48) := diag ( d , . . . , d i − , d i + d d , d i + , . . . , d d − ) . Then D (cid:48) ∈ D d − isinvertible, λ ( AD ) has a zero entry and λ ( A (cid:48) D (cid:48) ) is obtained from λ ( AD ) by deleting a zero entry . Proof
For each integer (cid:54) k (cid:54) d let s k denote the k -th row of A . Byassumption, s d = s i . Let λ be an eigenvalue of A (cid:48) D (cid:48) and let x (cid:48) := ( x , . . . , x d − ) ∈ R d − be acorresponding eigenvector. We define x := ( x , . . . , x d − , x i ) . For all (cid:54) k Proof of Theorem 1.2 We set Φ n := sup d (cid:62) max (cid:14) n (cid:88) k = λ k ( A ) : A ∈ A d (cid:15) . First, we show for all d (cid:62) n that Π ( n, d ) (cid:54) Φ n . We abbreviate π n ( AD ) := n (cid:88) k = λ k ( AD ) . Due to Theorem 2.1, there exist matrices A ∈ A d and D ∈ D d such that Π ( n, d ) = π n ( AD ) . Choose a sequence D k ∈ D d of invertible matrices with rational entriessatisfying Π ( n, d ) (cid:54) π n ( AD k ) + k . (2.2)This is possible since π n ( AD ) = π n ( √ DA √ D ) and because the map π n ( · ) is continuous on the set of symmetric matrices, cf. [OW92, p. 44]. Fix k (cid:62) .By finding a common denominator, we may write D k = diag ( n , . . . , n d ) , n i (cid:62) for all (cid:54) i (cid:54) d and m = n + · · · + n d . We set A k := bl ( n d − ) d ( · · · ( bl ( n − ) ( A )) · · · ) , where we use the convention bl ( A ) = A . Note that A k ∈ A m . By applyingLemma 2.2 repeatedly, we get that λ ( AD k ) is obtained from λ (cid:0) A k 1m m (cid:1) bydeleting exactly ( m − d ) zero entries. As a result, π n ( AD k ) (cid:54) π n ( A k ) m (cid:54) Φ n . (2.3)Thus, by combining (2.3) with (2.2), we obtain Π ( n, d ) (cid:54) Φ n . It is well-known that Π n = lim d → + ∞ Π ( n, d ) . Hence, Π n (cid:54) Φ n . The inequality Φ n (cid:54) Π n is a direct consequence of Theorem 2.1. Puttingeverything together, we conclude Π n = Φ n . We are left to show that it suffices to consider K n + -free two-graphs. Tothis end, fix an integer d > n and let A ∈ A d be a matrix such that π n ( A ) = max (cid:8) π n ( A (cid:48) ) : A (cid:48) ∈ A d (cid:9) . As the symmetric matrix A is orthogonally diagonalizable, there are or-thonormal vectors u , . . . , u n ∈ R d such that π n ( A ) = tr ( AUU t ) , where U is the matrix that has the vectors u i as columns. Let r k for (cid:54) k (cid:54) d be the rows of the matrix U . We use e , . . . , e d ∈ R d to denote the standardbasis. Fix (cid:54) i, j (cid:54) d and let ε ∈ R be a real number. We set A ( i, j ; ε ) := (cid:14) ε sgn (cid:0) (cid:104) r i , r j (cid:105) R n (cid:1) e i e tj if (cid:104) r i , r j (cid:105) (cid:54) = 0ε e i e tj otherwiseand (cid:98) A ε := A + (cid:0) A ( i, j ; ε ) + A ( j, i ; ε ) (cid:1) . Clearly, (cid:98) A ε is symmetric. Hereafter, we show that (cid:104) r i , r j (cid:105) (cid:54) = . To this end,suppose that (cid:104) r i , r j (cid:105) = 8e set ε (cid:63) := − sgn ( a ij ) , and we observe that (cid:98) A ε (cid:63) ∈ A d . Further, weabbreviate (cid:98) A := (cid:98) A ε (cid:63) . It holds that π n ( A ) = tr (cid:0) AUU t (cid:1) = tr (cid:16) (cid:98) AUU t (cid:17) − ε (cid:63) sgn (cid:0) (cid:104) r i , r j (cid:105) (cid:1) (cid:104) r i , r j (cid:105) . (2.4)Via von Neumann’s trace inequality, cf. [Mir75], we obtaintr (cid:16) (cid:98) AUU t (cid:17) (cid:54) π n ( (cid:98) A ) (cid:54) π n ( A ) ;thus, tr (cid:16) (cid:98) AUU t (cid:17) = π n ( (cid:98) A ) = π n ( A ) . The equality case of von Neumann’s trace inequality occurs. Therefore, thediagonalizable matrices UU t and (cid:98) A are simultaneously orthogonally diag-onalizable and thereby commute. This implies that UU t and (cid:0) A ( i, j ; ε (cid:63) ) + A ( j, i ; ε (cid:63) ) (cid:1) commute; as a result, we get that (cid:104) r i , r i (cid:105) = (cid:104) r j , r j (cid:105) , (cid:104) r i , r k (cid:105) = for all k (cid:54) = i with k ∈ { 1, . . . d } , (cid:104) r j , r k (cid:105) = for all k (cid:54) = j with k ∈ { 1, . . . d } . By applying the same argument to (cid:104) r i , r k (cid:105) = for every k (cid:54) = i, k ∈ { 1, . . . , d } , we may conclude that the vectors r , . . . , r d ∈ R n are orthogonaland none of them is equal to the zero vector. However, this is only possibleif n = d . Therefore, we have shown for d > n that (cid:104) r i , r j (cid:105) (cid:54) = for all integers (cid:54) i, j (cid:54) d .We claim that a ij = sgn (cid:0) (cid:104) r i , r j (cid:105) R n (cid:1) (2.5)for all (cid:54) i, j (cid:54) d . Because (cid:104) r i , r j (cid:105) (cid:54) = , this is a direct consequence of themaximality of π n ( A ) and equality (2.4). Hence, we have shown that A and UU t have the same sign pattern, which allows us to invoke [CW13, Lemma2.1]. From this result we see that A does not have a principal ( n + ) × ( n + ) -submatrix which has only − as off-diagonal elements. Such a matrix is theSeidel adjacency matrix of the complete graph on n + vertices. For thatreason, we have shown that 1d π n ( A ) = max (cid:14) nd + n (cid:88) k = λ k ( T ) : T is a K n + -free two-graph of order d (cid:15) . This completes the proof. (cid:4) We conclude this section with the proof of Corollary 1.3.9 roof of Corollary 1.3 Let J ∈ A denote the all-ones matrix. For every A ∈ A d , the matrix A ⊗ J is contained in A , where ⊗ denotes the Kro-necker product of matrices. Moreover, since the eigenvalues of A ⊗ J areprecisely all possible products of an eigenvalue of A (counted with mul-tiplicity) and an eigenvalue of J (counted with multiplicity), it is readilyverified that π n ( A ) d = π n ( A ⊗ J ) 2d . Let ( ε (cid:96) ) (cid:96) (cid:62) be a sequence of positive real numbers that converges to zero.Due to Theorem 1.2 and the above, there exists a strictly increasing sequence ( d (cid:96) ) (cid:96) (cid:62) of integers and matrices A (cid:96) ∈ A d (cid:96) such that Π n (cid:54) π n ( A (cid:96) ) d (cid:96) + ε (cid:96) . We have π n ( A (cid:96) ) = d (cid:96) − d (cid:96) (cid:88) k = n + λ k ( A (cid:96) ) = d (cid:96) + d (cid:96) − n (cid:88) k = λ k (− A (cid:96) ) ;thus, π n ( A (cid:96) ) = d (cid:96) + d (cid:96) − n (cid:88) k = λ k ( A (cid:96) ) − ( d (cid:96) − n ) where A (cid:96) = d (cid:96) − A (cid:96) . Consequently, Π n (cid:54) (cid:96) − + π d (cid:96) − n ( A (cid:96) ) d (cid:96) + ε (cid:96) . Since A (cid:96) ∈ A d (cid:96) , we obtain Π n (cid:54) (cid:96) − + Π ( d (cid:96) − n, d (cid:96) ) + ε (cid:96) . Proposition 2 in [FS17] tells us that Π ( d − n, d ) (cid:54) Π n + for all d (cid:62) . Thus, Π n (cid:54) (cid:96) − + Π ( d (cid:96) − n, d ) + ε (cid:96) (cid:54) Π n + (cid:96) + ε (cid:96) ;for that reason, the desired result follows. (cid:4) Polyhedral maximizers of Π n ( · ) E and E ∗ Let ( E, (cid:107)·(cid:107) ) be a Banach space and let V ⊂ E and F ⊂ E ∗ denote linear subspaces. We set V := (cid:8) (cid:96) ∈ E ∗ : (cid:96) ( v ) = for all v ∈ V (cid:9) ⊂ E and F := (cid:8) x ∈ E : f ( x ) = for all f ∈ F (cid:9) ⊂ E ∗ . Suppose that U ⊂ E is a linear subspace such that E = V ⊕ U . The map P UV : E → V, v + u (cid:55)→ v is a linear projection onto V . In the subsequent lemma we gather usefulresults from functional analysis. Lemma 3.1 Let ( E, (cid:107)·(cid:107) ) be a Banach space.1. If there exist closed linear subspaces V, U ⊂ E such that V is finite-dimensionaland E = V ⊕ U , then E ∗ = V ⊕ U , dim ( U ) = dim ( V ) , ( V ) = V and ( U ) = U. 2. If there exist closed linear subspaces F, G ⊂ E ∗ such that F is finite-dimensionaland E ∗ = F ⊕ G , then E = F ⊕ G , dim ( G ) = dim ( F ) , ( F ) = F and ( G ) = G. 3. If there exist closed linear subspaces V, U ⊂ E such that V is finite-dimensionaland E = V ⊕ U , then (cid:107) P UV (cid:107) = (cid:107) P V U (cid:107) . Proof We prove each item separately.1. If (cid:96) ∈ V ∩ U , then (cid:96) ( x ) = for all x ∈ E , implying V ∩ U = { } .As V and U are closed, we may deduce with the usual Hahn-Banachseparation argument that ( V ) = V and ( U ) = U. Accordingly, (( V ) ) = V and (( U ) ) = U . So, V and U areweak-star closed, cf. [Rud91, Theorem 4.7]. The quotient E ∗ /V is aHausdorff locally convex vector space and the quotient map π : E ∗ → E ∗ /V is continuous. We claim that the subspace π ( U + V ) is closedin E ∗ /V . As every finite-dimensional linear subspace of a Hausdorff11opological vector space is closed, cf. [Rud91, Theorem 1.21], it sufficesto show that π ( U + V ) is finite-dimensional. Since ( E/U ) ∗ = U , we see that U is finite-dimensional; thus, the subspace π ( U + V ) is finite-dimensional as well. Therefore, the subspace π ( U + V ) isclosed in E ∗ /V and we get that V + U = π − ( π ( U + V )) is weak-star closed. Note that V + U is weak-star dense in E ∗ , because V + U separates points of E . This implies E ∗ = V ⊕ U , as desired.2. Suppose that x ∈ F ∩ G . We have (cid:96) ( x ) = for all (cid:96) ∈ E ∗ . As theelements of E ∗ separate points, we get that x = and consequently F ∩ G = { } . Since F is finite-dimensional and thereby weak-star closed,[Rud91, Theorem 4.7] tells us that ( F ) = F. Using ( E/F ) ∗ = ( F ) = F, we may deduce that E/F is finite-dimensional. Let π : E → E/F de-note the quotient map. The linear subspace π ( G + F ) ⊂ E/F is finite-dimensional and thus closed. As a result, F + G = π − ( π ( G + F )) is closed. Via the familiar Hahn-Banach separation argument, it is nothard to check that F + G is a dense subset of E . For that reason, F ⊕ G = E . The first item tells us that ( F ) ⊕ ( G ) = E ∗ ;therefore ( G ) = G , as ( F ) = F . This completes the proof of thesecond item.3. We abbreviate F := V and G := U . We compute (cid:107) P V U (cid:107) = sup (cid:107) f + g (cid:107) = ∈ F, g ∈ G (cid:107) g (cid:107) = sup (cid:107) f + g (cid:107) = ∈ F, g ∈ G sup (cid:107) v + u (cid:107) = ∈ V, u ∈ U | g ( v ) | and sup (cid:107) v + u (cid:107) = ∈ V, u ∈ U sup (cid:107) f + g (cid:107) = ∈ F, g ∈ G | g ( v ) | = sup (cid:107) v + u (cid:107) = ∈ V, u ∈ U (cid:107) v (cid:107) = (cid:107) P UV (cid:107) , as was to be shown. (cid:4) .2 Construction of polyhedral maximizers Let ( E, (cid:107)·(cid:107) ) be a Banachspace and let F ⊂ E denote a finite-dimensional linear subspace. The number Π ( F, E ) := inf (cid:8) (cid:107) P (cid:107) | P : E → F bounded surjective linear map with P = P (cid:9) . is called the relative projection constant of F with respect to E . The followingtheorem translates the calculation of relative projection constants to secondpreduals (if such a space exists). Theorem 3.2 Let ( E, (cid:107)·(cid:107) ) be a Banach space and let F ⊂ E denote a finite-dimensional linear subspace. If ( X, (cid:107)·(cid:107) ) is a Banach space such that E = X ∗∗ , thenthere exist a linear subspace V ⊂ X with dim ( V ) = dim ( F ) and Π ( F, E ) = Π ( V, X ) . Proof It is not hard to check that Π ( F, E ) := inf (cid:8) (cid:107) P GF (cid:107) : E = F ⊕ G, G ⊂ E closed linear subspace (cid:9) . We set V := ( F ) . On the one hand, using the second and third item ofLemma 3.1, we obtain Π ( V, X ) (cid:54) Π ( F, E ) ;on the other hand, using the first and third item of Lemma 3.1, we infer Π ( F, E ) (cid:54) Π ( V, X ) . This completes the proof. (cid:4) We conclude this section with the proof of Theorem 1.4. Proof of Theorem 1.4 Let F ⊂ (cid:96) ∞ be an n -dimensional linear subspacewith Π n = Π ( F, (cid:96) ∞ ) . Via Theorem 3.2, there exists an n -dimensional linear subspace V ⊂ c suchthat Π ( F, (cid:96) ∞ ) = Π ( V, c ) . As Π ( V, c ) (cid:54) Π ( V ) , we get Π n = Π ( V ) . This completes the proof, since due to a result of Klee, cf. [Kle60, Proposition4.7], every finite-dimensional subspace of c is polyhedral. (cid:4) Applications: Computation of Π and Π ( 4, 6 ) K -free two-graphs Let n (cid:62) be an integer, let R + ⊂ R be a regular ( + ) -gon centred at the origin and let V ( R + ) denotethe vertices of R + . Further, we let T + denote the two-graph that has V ( R + ) as vertex set and { v , v , v } ⊂ V ( R + ) is an edge if and onlyif the origin is contained in the closed convex hull of v , v , v . It is readilyverified that δ ( R + − + ) = T + for R + := t − j t j J n J n − n − j J n − tn J n , where j ∈ R n is the all-ones vector, J n is the all-ones n × n -matrix and L n isthe n × n -matrix given by ( L n ) ij := (cid:14) − otherwise . Note that L n has only − ’s below the diagonal and only ’s above the firstsub-diagonal. We set A := − − 11 1 1 − − 11 1 − − − − − − . One can check that A − is the Seidel adjacency matrix of the graphdepicted in Figure 1. We abbreviate Ω := (cid:8) S : S is a principal submatrix of A − (cid:9) ∪ (cid:8) R + − + : n (cid:62) (cid:9) . In [FF84], Frankl and Füredi showed that each non-empty K -free two-graphbelongs to the set δ ( Ω ) ∪ (cid:8) δ ( B ) : B is a blow-up of a matrix in Ω (cid:9) . n eigenvalues of AD ? Given a matrix A ∈ A d , we denote by Stab ( A ) the set (cid:8) Q ∈ O d ( Z ) : A = QAQ t (cid:9) . We use O d ( Z ) to denote the group of orthogonal d × d -matrices with integerentries. Every Q ∈ Stab ( A ) has a unique decomposition Q = PD , where P 23 4 56 Figure 1: The graph that has A − as Seidel adjacency matrix.is a permutation matrix and D is a diagonal matrix consisting only of ’sand − ’s. We write P τ := P if the permutation matrix P is associated to thepermutation τ , that is, P ij = ( e τ ( i ) ) j . The group Stab ( A ) acts on { 1, . . . , d } via ( P τ D, k ) (cid:55)→ τ ( k ) . Two Seidel adjacency matrices S f and S g are called switching equivalent if δ ( f ) = δ ( g ) . This gives rise to an equivalence relation, equivalence classesare called switching classes. The lemma below tells us that the orbit decom-position of the action Stab ( A ) (cid:121) { 1, . . . , d } may be obtained by determiningthe switching class of every principal ( d − )− dimensional submatrix of A . Lemma 4.1 Let A ∈ A d be a matrix, let (cid:54) i, j (cid:54) d be two integers and for k = i, j let T k denote the submatrix of A − d obtained by deleting the k -th columnand the k -th row of A − d .Then, the matrices T i and T j are switching equivalent if and only if the integers i and j lie in the same orbit under the action Stab ( A ) (cid:121) { 1, . . . , d } . Proof This is a straightforward consequence of the definitions. (cid:4) Let M be a diagonalizable d × d -matrix over the real numbers. We set π n ( M ) := n (cid:88) k = λ k ( M ) for each integer (cid:54) n (cid:54) d . The following lemma simplifies the calculationof the maximum value of the function D (cid:55)→ π n ( AD ) if the action Stab ( A ) (cid:121) { 1, . . . , d } is transitive. 15 emma 4.2 Let A ∈ A d be a matrix and let (cid:54) d be an integer. If Λ ∈ D d is a invertible matrix such that π n ( AΛ ) = max D ∈ D d π n ( AD ) , then Q Λ ( Q ) t = Λ for all Q ∈ Stab ( A ) . In particular, if d is odd and the action Stab ( A ) (cid:121) { 1, . . . , d } is transitive, then Λ = d . Proof For each Q ∈ Stab ( A ) we have Q t √ ΛA √ ΛQ = Q t √ ΛQAQ t √ ΛQ = (cid:113) Λ Q t A (cid:112) Λ Q , where Λ Q := QλQ t . Consequently, π n ( √ ΛA √ Λ ) = π n (cid:16)(cid:113) Λ Q t A (cid:112) Λ Q (cid:17) = π n (cid:16) A (cid:112) Λ Q (cid:113) Λ Q t (cid:17) . We get (cid:54) tr (cid:16)(cid:112) Λ Q (cid:113) Λ Q t (cid:17) . Via the Cauchy-Schwarz inequality, we deducetr (cid:16)(cid:112) Λ Q (cid:113) Λ Q t (cid:17) (cid:54) ;as a result, there exists a real number α (cid:62) such that Λ Q = αΛ Q t . Since tr ( Λ Q ) = tr ( Λ Q t ) = , we get α = and thus Λ Q = Λ Q t , which is equivalent to Λ Q = Λ. Now, suppose that d is odd and assume that the action Stab ( A ) (cid:121) { 1, . . . , d } is transitive. We claim that Λ = d . The statement follows via elementarygroup theory. Indeed, let H denote the subgroup of Stab ( A ) generated bythe squares. By basic algebra, H is normal and the action of Stab ( A ) /H onthe orbits of H (cid:121) { 1, . . . , d } is transitive. Since | Stab ( A ) /H | = k for someinteger k (cid:62) , the action H (cid:121) { 1, . . . , d } has either one orbit or an evennumber of orbits. Because d is odd and the orbits of H (cid:121) { 1, . . . , d } all havethe same cardinality, we may conclude that H (cid:121) { 1, . . . , d } is transitive. Thiscompletes the proof. (cid:4) .3 Determination of Π In the following we retain the notation fromParagraph 4.1. By the use of Theorem 1.2, Lemma 2.2 and the classificationof all K -free two-graphs, we obtain Π = max ( A − ) ∈ Ω max D ∈ D d π ( AD ) . Clearly, all induced sub-graphs of T + that are obtained by deleting onevertex are isomorphic (as two-graphs) to each other. Thus, via Lemma 4.1and Lemma 4.2, we get thatmax D ∈ D d π ( R + D ) = π (cid:0) + R + (cid:1) . Moreover, if B is a principal submatrix of A , then it is not hard to see thatmax D ∈ D d π ( BD ) (cid:54) max (cid:8) π (cid:0) R (cid:1) , π (cid:0) R (cid:1) (cid:9) ;thereby, Π = max n (cid:62) π (cid:0) + R + (cid:1) . Thus, we are left to consider the eigenvalues of the matrices R + for n (cid:62) .Due to the following lemma it suffices to calculate the eigenvalues of R . Lemma 4.3 Let n (cid:48) (cid:62) n (cid:62) be integers. It holds π (cid:0) + R + (cid:1) (cid:62) π (cid:0) (cid:48) + R (cid:48) + (cid:1) (4.1) Proof We abbreviate N := + . Let R (cid:48) N denote the × -matrix that isobtained from R N by deleting the second row and second column. Clearly, R (cid:48) N is a blow-up of R N − ; thus, via Lemma 2.2, we obtainmax D ∈ D π ( R (cid:48) N D ) (cid:54) π ( R N − ) . If for all integers k (cid:62) ( R + ) = ( R + ) , (4.2)then π (cid:0) R N 1N (cid:1) = (cid:0) R (cid:48) N 1N − (cid:1) N − (cid:54) (cid:0) R N − − (cid:1) N − and thus (4.1) follows. We are left to show that (4.2) holds.Suppose that λ ( R N ) has multiplicity one. Below, we show that this leadsto a contradiction.Let x ∈ R N be an eigenvector of R N associated to the eigenvalue λ ( R N ) .As we assume that λ ( R N ) has multiplicity one, we get Qx = x or Qx = − x for each Q ∈ Stab ( R N ) . We know that the action Stab ( R N ) (cid:121) { 1, . . . , N } istransitive; thus all entries of x differ only by a sign. Without loss of generality17e may suppose the entries of x consist only of ’s and − ’s. For each integer (cid:54) i (cid:54) N let A i denote the matrix that is obtained from R N by replacingthe i -th column with x . Cramers rule tells us that x i det ( R N ) = det ( A i ) for all (cid:54) i (cid:54) N . It is easy to see (via the definition of R N ) that for all n < i < N : if x i − n + and x i − n have the same sign, then det ( A i ) = . But thisis impossible; for that reason, for all n < i < N we have x i − n = − x i − n + .Similarly, x i + n = x i + n − for all (cid:54) n + and x = − x n + , x = − x N . Thus, if we suppose that x = , then x = ( − 1, 1, − 1, 1, . . . 1, − (cid:124) (cid:123)(cid:122) (cid:125) n times , 1, − 1, 1, − 1, . . . , − 1, 1 (cid:124) (cid:123)(cid:122) (cid:125) n times ) if n is oddand x = ( 1, 1, − 1, 1, − 1, . . . , 1, − (cid:124) (cid:123)(cid:122) (cid:125) n times , 1, − 1, 1, − 1, . . . , 1, − (cid:124) (cid:123)(cid:122) (cid:125) n times ) if n is even . Therefore, if j ∈ R N denotes the all-ones vector we obtain (cid:104) x, j (cid:105) = and consequently it holds that λ ( R N ) = (cid:14) − if n is odd if n is even . This is a contradiction, since tr ( R N ) = N and we assume that λ ( R N ) hasmultiplicity one. Hence, we have shown that the eigenvalue λ ( R N ) has mul-tiplicity greater than or equal to two. As a result, (4.2) holds, which was leftto show. This completes the proof. (cid:4) Employing Lemma 4.3, we get Π = π (cid:0) R (cid:1) = (cid:0) ( R ) (cid:1) = (cid:0) − λ ( R ) (cid:1) = 43 , as conjectured by Grünbaum. 18 .4 An illustrative example: Π ( 4, 6 ) = . In this paragraph, we showthat Π ( 4, 6 ) = . We hope that some of the tools that are developed heremay also simplify the computation of other relative projection constants.From a result of Sokołowski, cf. [Sok17], we obtain (cid:54) Π ( 4, 6 ) . Given A ∈ A , we let n + ( A ) (or n − ( A ) ) denote the number of positive (or negative)eigenvalues of A counted with multiplicity. Since Π ( 3, 6 ) < and Π ( 5, 6 ) < , we deduce with the help of Theorem 2.1 that Π ( 4, 6 ) = max (cid:8) π ( AD ) : A ∈ A , n + ( A ) = 4, n − ( A ) = 2, D ∈ D (cid:9) . In [BMS81], Bussemaker, Mathon and Seidel classified all two-graphs on sixvertices. Using this classification, we get that there are exactly three non-isomorphic two-graphs on six vertices with signature ( n + , n − ) = ( 4, 2 ) ,namely T i := [ A i ] , i = 1, 2, 3, for A := − − − 11 1 1 1 − − − 11 1 − − ,A := − − − − 11 1 1 − − 11 1 1 − − , and A := − 11 1 1 1 − − − − − . Therefore, Π ( 4, 6 ) = max i = M i , for M i := max D ∈ D π ( A i D ) . In [LS91], Lieb and Siedentop established that the function D ∈ D d (cid:55)→ π n ( AD ) is concave if A is invertible and n + ( A ) = n . This result and the followinglemma simplifies the computation of Π ( 4, 6 ) .19 emma 4.4 Let A ∈ A d be an invertible matrix that has n := n + ( A ) positiveeigenvalues. If the supremum M := sup (cid:8) π n ( AD ) : D ∈ D d is invertible (cid:9) is attained, then there exist an invertible matrix Λ ∈ D d such that π n ( AΛ ) = M and QΛQ t = Λ for all Q ∈ Stab ( A ) . (4.3) Proof We set X = (cid:8) D ∈ D d : D is invertible and π n ( AD ) = M (cid:9) . The function π n ( AD ) is concave if we restrict the source space to the non-singular matrices in D d ; accordingly, it is immediate that the subset X ⊂ D n is convex. Let Q ∈ Stab ( A ) . For all D ∈ D d we obtain π n ( AD ) = π n ( Q t AQD ) = π n ( QQ t AQDQ t ) = π n ( AQDQ t ) . for that reason, for all Q ∈ Stab ( A ) the set X is invariant under conjugationwith Q . If we equip X with the euclidean distance (cid:107)·(cid:107) , then every map x ∈ X (cid:55)→ QxQ t is an isometry. Since ( X, (cid:107)·(cid:107) ) is a bounded non-emptyCAT(0) space, there is a matrix Λ ∈ X ⊂ D d with the desired properties, cf.[BH13, II. Corollary 2.8]. This completes the proof. (cid:4) To begin, we show that M (cid:54) 53 . (4.4)The value M will be estimated afterwards. By drawing the underlyinggraph of A and employing Lemma 4.1, we may deduce that the actionStab ( A ) (cid:121) { 1, . . . , 6 } has orbit decomposition { 1, 2 } , { 3, 4 } , { 5, 6 } ; consequently,Lemma 4.4 tells us that M = max (cid:8) π (cid:0) A Λ ( s, t, w ) (cid:1) : ( s, t, w ) ∈ ∆ (cid:9) , where ∆ ⊂ R is the 2-dimensional standard simplex and Λ ( s, t, w ) := diag ( s/2, s/2, t/2, t/2, w/2, w/2 ) .The characteristic polynomial p ( x ) of the matrix A Λ ( s, t, w ) can bewritten as p ( x ) = q ( x ) q ( x ) q ( x ) , for q ( x ) := x − s,q ( x ) := − x + wx + tw and q ( x ) := x − tx − s ( − s ) x + stw. q ( x ) are (cid:16) w ± (cid:112) w + (cid:17) . As p ( x ) has four positive roots, we obtain that q ( x ) has two positive roots.Let x (cid:62) x (cid:62) x denote the roots of q . We need to bound x + x fromabove. By the virtue of Vieta’s formulas, the roots ξ (cid:62) ξ (cid:62) ξ of thepolynomial h ( x ) := x − + (cid:0) t − s ( − s ) (cid:1) x + st satisfy ( ξ , ξ , ξ ) = ( x + x , x + x , x + x ) . Hence, in order to estimate x + x from above, it suffices to bound the largest root of the polynomial h from above. In the subsequent lemma we use Taylor’s Theorem to get anupper bound for the largest root of a cubic polynomial with three real roots. Lemma 4.5 Let p ( x ) = x + bx + cx + d , with b, c, d ∈ R , be a polynomial. If p has three real roots x (cid:62) x (cid:62) x , then x (cid:54) (cid:18) − b + √ √ − A + C6 (− A ) (cid:19) , for A := − b , − C := − + Proof As all roots are real, the cubic formula tells us that x = − b3 + √ − A cos (cid:32) arccos (cid:32) C2 (− A ) (cid:33)(cid:33) . For a > 0 we define the map h a : [− , 2a ] → R via h a ( x ) := a cos (cid:18) arccos (cid:16) x2a (cid:17)(cid:19) . From Taylor’s Theorem for each x ∈ [− , 2a ] it holds that h a ( x ) = √ + sin (cid:0) arccos (cid:0) c2a (cid:1)(cid:1) (cid:113) − c x, where c is a real number between and x . Using elementary analysis, weobtain (cid:14) sin (cid:0) arccos ( x ) (cid:1) (cid:54) √ − x for all x ∈ [ 0, 1 ]− sin (cid:0) arccos ( x ) (cid:1) (cid:54) − √ − x for all x ∈ [− 1, 0 ] . (cid:0) arccos (cid:0) c2a (cid:1)(cid:1) (cid:113) − c x (cid:54) x12a (4.5)for all x ∈ [− , 2a ] . All things considered, we get x (cid:54) (cid:18) − b + √ √ − A + C6 (− A ) (cid:19) , as was to be shown. (cid:4) Employing the splitting p ( x ) = q ( x ) q ( x ) q ( x ) and Lemma 4.5, weinfer that π (cid:0) A Λ (cid:1) is less than or equal to f ( s, t, w ) := s + (cid:16) w + (cid:112) w + (cid:17) + (cid:18) + √ √ − A + C6 (− A ) (cid:19) , where A := − + − t , C := − + ( − s ) st − . By elementary analysis, C6 (− A ) (cid:54) s + w2 ;as a result, we obtain f ( s, t, w ) (cid:54) + (cid:114) w + tw + (cid:114) t + s ( − s ) . Clearly, (cid:114) w + tw + (cid:114) t + s ( − s ) (cid:54) (cid:114) w + tw + (cid:114) t + s ( − s ) . (4.6)Via Lagrange multipliers, the maximal value over ∆ of the right hand sideof (4.6) is equal to (cid:115) (cid:16) − √ (cid:17) + − √ √ + = So, M (cid:54) 53 , as desired. 22ext, we proceed with exactly the same strategy that dealt with M toshow that M (cid:54) 53 . As before, one can verify that the action Stab ( A ) (cid:121) { 1, . . . , 6 } has orbit de-composition { } , { 2, 3 } , { 4, 5, 6 } . The characteristic polynomial of A Λ ( s, t, w ) ,for Λ ( s, t, w ) := diag ( s, t/2, t/2, w/3, w/3, w/3 ) , is given by p ( x ) = − r ( x ) r ( x ) r ( x ) , with r ( x ) := t − x,r ( x ) := ( − ) , and r ( x ) := x − (cid:16) s − w3 (cid:17) x − (cid:18) st + 43 sw + tw (cid:19) x − 43 stw. Since all the roots of r ( x ) and r ( x ) are positive, we see that r ( x ) has twonegative roots. Thus, employing Lemma 4.5 and Lemma 4.4, we obtain that π (cid:0) A Λ ( s, t, w ) (cid:1) is less than or equal to g ( s, t, w ) := t + 43 w + (cid:18) s − w3 + √ √ − A + C6 (− A ) (cid:19) , where A := − − ( w − ) − − := + t + w + − − − 27 . By elementary analysis, it is possible to verify that C18 (− A ) (cid:54) 14 s + (cid:16) √ − (cid:17) t ;for that reason, g ( s, t, w ) (cid:54) + 14 s + (cid:16) √ − (cid:17) t + 89 w + (cid:114) + + + + w Via Lagrange multipliers, the maximal value over ∆ of the right hand sideof the above is equal to for s := 0, t := (cid:16) √ + (cid:17) := − (cid:16) √ − (cid:17) M (cid:54) 53 . Next, we show that M = . The action Stab ( A ) (cid:121) { 1, . . . , 6 } is transitive,that is, has orbit decomposition { 1, 2, 4, 5, 6 } . Thus, by Lemma 4.4, we deducethat M = π ( A ) . Let J denote the × -dimensional all-ones matrix and let R denote thematrix introduced in Paragraph 4.1. Since π (cid:0) R (cid:1) = , π (cid:0) J ⊗ R (cid:1) = 43 , where we use ⊗ to denote the Kronecker product of matrices. Consequently,for J ⊗ R = − J ⊗ R , we get π (cid:0) J ⊗ R (cid:1) = + π (cid:0) J k ⊗ R (cid:1) − = − = 53 . Considering J ⊗ R = A , we get M = and finally Π ( 4, 6 ) = 53 , as claimed. Remark 4.6 As R admits a system of ten equiangular lines, cf. [LS73, p.496], the general upper bound of König, Lewis, and Lin, cf. [KLL83], tells usthat Π ( 5, 10 ) = To summarize, we obtain the sequence Π ( 1, 1 ) = 1, Π ( 2, 3 ) = 43 , Π ( 4, 6 ) = 53 , Π ( 5, 10 ) = which naturally leads to the open question: Are there integers n (cid:54) d suchthat Π ( n, d ) = ? I am thankful to Urs Lang who read earlierdraft versions of this article and who stimulated several improvements.Moreover, I am grateful to Anna Bot for proofreading this paper.24 eferences [Ban32] S. Banach. “Théorie des opérations linéaires”. In: Instytut Matem-atyczny Polskiej Akademi Nauk (1932).[BH13] M. R. Bridson and A. Haefliger. Metric spaces of non-positive curva-ture . Vol. 319. Springer Science & Business Media, 2013.[BMS81] F. Bussemaker, R. Mathon, and J. J. Seidel. “Tables of two-graphs”.In: Combinatorics and graph theory . Springer, 1981, pp. 70–112.[Boh38] F. Bohnenblust. “Convex regions and projections in Minkowskispaces”. In: Annals of Mathematics (1938), pp. 301–308.[CL09] B. L. Chalmers and G. Lewicki. “Three-dimensional subspace of (cid:96) ∞ with maximal projection constant”. In: Journal of FunctionalAnalysis Studia Mathematica Electronic Journal of LinearAlgebra Discretemathematics 50 (1984), pp. 323–328.[FS17] S. Foucart and L. Skrzypek. “On maximal relative projection con-stants”. In: Journal of Mathematical Analysis and Applications Transactions of the Amer-ican Mathematical Society North-Holland Mathematics Studies . Vol. 7. Elsevier, 1973, pp. 80–83.[Kle60] V. Klee. “Polyhedral sections of convex bodies”. In: Acta Math. Studia Mathematica Mat. Zametki Studia Mathematica Israel Jour-nal of Mathematics Michigan Math. J. Journal of Alge-bra Journal of Statistical Physics Monat-shefte für Mathematik SIAM Journal on MatrixAnalysis and Applications Contemporary Mathematics 414 (2006), p. 147.[Rud91] W. Rudin. Functional analysis. International series in pure and appliedmathematics . 1991.[Sei91] J. Seidel. “A survey of two-graphs”. In: Geometry and Combina-torics . Ed. by D. Corneil and R. Mathon. Academic Press, 1991,pp. 146 –176.[Sei92] J. Seidel. “More About Two-Graphs”. In: Fourth CzechoslovakianSymposium on Combinatorics, Graphs and Complexity . Ed. by J. Neˆsetriland M. Fiedler. Vol. 51. Annals of Discrete Mathematics. Elsevier,1992, pp. 297 –307.[Sok17] F. Sokołowski. “Minimal Projections onto Subspaces with Codi-mension 2”. In: Numerical Functional Analysis and Optimization Proceedings of the LondonMathematical Society J. Res. Nat. Bur.Standards Sect. B Banach-Mazur distances and finite-dimensionaloperator ideals . Vol. 38. Longman Scientific & Technical, 1989. M athematik D epartement , ETH Z ürich , R ämistrasse ürich , S chweiz E-mail adress: [email protected]@math.ethz.ch