The Viterbo's capacity conjectures for convex toric domains and the product of a 1 -unconditional convex body and its polar
aa r X i v : . [ m a t h . S G ] A ug The Viterbo’s capacity conjectures for convex toric domains andthe product of a 1-unconditional convex body and its polar
Kun Shi and Guangcun Lu ∗ August 20, 2020
Abstract
In this note, we show that the strong Viterbo conjecture holds true on any convex toricdomain, and that the Viterbo’s volume-capacity conjecture holds for the product of a 1-unconditional convex body A ⊂ R n and its polar. We also give a direct calculus proof of thesymmetric Mahler conjecture for l p -balls. Prompted by Gromov’s seminal work [7] Ekeland and Hofer [5] defined a symplectic capacity on the 2 n -dimensional Euclidean space R n with the standard symplectic structure ω to bea map c which associates to each subset U ⊂ R n a number a number c ( U ) ∈ [0 , ∞ ] satisfyingthe following axioms: (Monotonicity) c ( U ) ≤ c ( V ) for U ⊂ V ⊂ R n ; (Conformality) c ( ψ ( U )) = | α | c ( U ) for ψ ∈ Diff( R n ) such that ψ ∗ ω = αω with α = 0; (Nontriviality) < c ( B n (1)) and c ( Z n (1)) < ∞ , where B n ( r ) = { z ∈ R n | | z | < r } and Z n ( R ) = B ( R ) × R n − .Moreover, such a symplectic capacity is called normalized if it also satisfies (Normalization) c ( B n (1)) = c ( Z n (1)) = π .(Without special statements we make conventions : 1) symplectic capacities on R n are allconcerning the symplectic structure ω ; 2) a “ domain ” in a Euclidean space always denotes theclosure of an open subset; 3) the notation h· , ·i always denotes the Euclidean inner product.)Hofer and Zehnder [12] extended the concept of a symplectic capacity to general symplecticmanifolds. The first example of a normalized symplectic capacity is the Gromov width w G ,which maps a 2 n -dimensional symplectic manifold ( M, ω ) to w G ( M, ω ) = sup { πr | ∃ a symplectic embedding ( B n ( r ) , ω ) ֒ → ( M, ω ) } . (1.1)In particular, for a subset U ⊂ R n it can be easily proved that w G ( U ) := w G ( U, ω ) = sup { πr | ∃ ψ ∈ Symp( R n ) with ψ ( B n ( r )) ⊂ U } ∗ Corresponding authorPartially supported by the NNSF 11271044 of China.2020
Mathematics Subject Classification. ith the Extension after Restriction Principle for symplectic embeddings of bounded star-shaped open domains (see Appendix A in [28]). Clearly c Z ( U ) := sup { πr | ∃ ψ ∈ Symp( R n ) with ψ ( U ) ⊂ Z n ( r )) } defines a normalized symplectic capacity on R n , the so-called cylindrical capacity . Nowadays,a variety of normalized symplectic capacities can be constructed in categories of symplecticmanifolds for the study of different problems, for example, the (first) Ekeland-Hofer capacity c EH ([5]), the Hofer-Zehnder capacity c HZ ([12]) and Hofer’s displacement energy e ([11]), theFloer-Hofer capacity c FH ([6]) and Viterbo’s generating function capacity c V ([32])), the firstGutt-Hutchings capacity c CH1 ([8]) coming from S -equivariant symplectic homology, and thefirst ECH capacity c ECH1 in dimension 4 ([13]). Except the last c ECH1 the others have definedfor all convex domains in ( R n , ω ). As an immediate consequence of the normalization axiomwe see that w G and c Z are the smallest and largest normalized symplectic capacities on R n ,respectively. An important open question in symplectic topology ([20, 19]), termed the strongViterbo conjecture ([9]), states that w G and c Z coincide on convex domains in R n , that is, Conjecture 1.1.
All normalized symplectic capacities coincide on convex domains in R n . Conjecture 1.2 (Viterbo [33]) . On R n , for any normalized symplectic capacity c and anybounded convex domain D there holds c ( D ) c ( B n (1)) ≤ (cid:18) Vol( D )Vol( B n (1)) (cid:19) /n (1.2)(or equivalently ( c ( D )) n ≤ Vol(
D, ω n ) = n !Vol( D )), with equality if and only if D is symplec-tomorphic to the Euclidean ball, where Vol( D ) denotes the Euclidean volume of D .Since (1.2) is clearly true for c = w G , Conjecture 1.2 follows from Conjecture 1.1. Somespecial cases of Conjecture 1.2 were proved in [2, 15].Surprisingly, Artstein-Avidan, Karasev, and Ostrover [1] showed that Conjecture 1.2 im-plies the following long-standing famous conjecture about the Mahler volume M (∆) := Vol(∆ × ∆ ◦ ) = Vol(∆)Vol(∆ ◦ )of a bounded convex domain ∆ ⊂ R n in convex geometry, where ∆ ◦ = { x ∈ R n | h y, x i ≤ ∀ y ∈ ∆ } is the polar of ∆. Conjecture 1.3 ( Symmetric Mahler conjecture [18]) . M (∆) ≥ n n ! for any centrallysymmetric bounded convex domain ∆ ⊂ R n .The n = 2 case of this conjecture was proved by Mahler [18]. Iriyeh and Shibata [14] havevery recently proved the n = 3 case. Some special classes of centrally symmetric boundedconvex domains in R n , for example, those with 1-unconditional basis, zonoids, polytopes withat most 2 n + 2 facets, were proved to satisfy Conjecture 1.3 in [30], [25] and [17], respectively.Karasev [16] recently confirmed the conjecture for hyperplane sections or projections of l p -ballsor the Hanner polytopes. See [29, 31] and the references of [14] for more information.Hermann [10] proved Conjecture 1.1 for convex Reinhardt domains D . Recall that a subset X of C n is called a Reinhardt domain ([10]) if it is invariant under the standard toric action T n = R n / Z n on C n defined by( θ , · · · , θ n ) · ( z , · · · , z n ) = (cid:0) e πiθ z , · · · , e πiθ z n (cid:1) . (1.3)This is a Hamiltonian action (with respect to the standard symplectic structure ω on C n = R n ) with the moment map µ : C n → R n , ( z , · · · , z n ) ( π | z | , · · · , π | z n | ) fter identifying the dual of the Lie algebra of T n with R n .Let R n ≥ (resp. Z n ≥ ) denote the set of x ∈ R n (resp. x ∈ Z n ) such that x i ≥ i = 1 , . . . , n . Given a nonempty relative open subset Ω in R n ≥ we call Reinhardt domains X Ω = µ − (Ω) and X Ω = µ − (Ω) toric domains associated to Ω and Ω (the closure of Ω), respectively. (Both X Ω and X Ω havevolumes Vol(Ω) by [10, Lemma 2.6].) Moreover, following [8], if Ω is bounded, and b Ω = { ( x , · · · , x n ) ∈ R n | ( | x | , · · · , | x n | ) ∈ Ω } (resp. R n ≥ \ Ω)is convex (resp. concave) in R n , we said X Ω and X Ω to be convex toric domains (resp. concavetoric domains ). There exists an equivalent definition in [24]. An open and bounded subset A ⊂ R n is called a balanced region if [ −| x | , | x | ] × · · · × [ −| x n | , | x n | ] ⊂ A for each ( x , · · · , x n ) ∈ A .Such a set A is determined by the relative open subset | A | := A ∩ R n ≥ in R n ≥ . For a nonemptyrelative open subset Ω in R n ≥ there exists a balanced region A ⊂ R n such that Ω = | A | if andonly if [0 , | x | ] × · · · × [0 , | x n | ] ⊂ Ω for each ( x , · · · , x n ) ∈ Ω ([24, Remark 10]). The balancedregion A ⊂ R n is said to be convex (resp. concave ) if A (resp. R n ≥ \ A ) is convex in R n . Then X | A | is convex (resp. concave ) in the sense above if and only if the balanced region A ⊂ R n isconvex (resp. concave). Clearly, the balanced regions are centrally symmetric, and any convexor concave balanced region is star-shaped. By [10, Lemma 2.5] each convex or concave toricdomains is star-shaped.By [8, Examples 1.5, 1.12], a 4-dimensional toric domain X Ω is convex (resp. concave) ifand only if Ω = { ( x , x ) | ≤ x ≤ a, ≤ x ≤ f ( x ) } (1.4)where f : [0 , a ] → R ≥ is a nonincreasing concave function (resp. convex function with f ( a ) = 0). ( Note that the concept of the present -dimensional convex toric domain is strongerthan one in [4].)Let X Ω be a convex or concave toric domain associated to Ω ⊂ R n ≥ as above, and let Σ Ω and Σ Ω be the closures of the sets ∂ Ω ∩ R n> and ∂ Ω ∩ R n> , respectively. (Clearly, Σ Ω = Σ Ω .)For v ∈ R n ≥ we define k v k ∗ Ω = sup {h v, w i | w ∈ Ω } = max {h v, w i | w ∈ Ω } = k v k ∗ Ω , (1.5)[ v ] Ω = min {h v, w i | w ∈ Σ Ω } = min {h v, w i | w ∈ Σ Ω } = [ v ] ∗ Ω (1.6)([8, (1.9) and (1.13)]). Then [ v ] Ω ≤ k v k ∗ Ω , and k v k ∗ r Ω = r k v k ∗ Ω and [ v ] r Ω = r [ v ] Ω for all r > c EH , c CH1 , c V and w G coincideon any convex or concave toric domain. Combing the latter assertion with a result in [8] wecan easily obtain the first result of this note, which claims that Conjecture 1.1 and thereforeConjecture 1.2 holds true on all convex toric domains in R n . More precisely, we have: Theorem 1.4.
Let Ω ⊂ R n ≥ be a bounded nonempty relative open subset such that b Ω is convexin R n . Then for any normalized symplectic capacity c on R n convex toric domains X Ω and X Ω have capacities c ( X Ω ) = c ( X Ω ) = min ( k v k ∗ Ω (cid:12)(cid:12)(cid:12) v = ( v , · · · , v n ) ∈ Z n ≥ , n X i =1 v i = 1 ) = min {k e i k ∗ Ω | i = 1 , · · · , n } , where { e i } ni =1 is the standard orthogonal basis of R n . t is unclear whether convex toric domains must be convex Reinhardt domains in R n . Butthe following Corollary 1.6 shows that Conjecture 1.1 holds true for a class of convex domainsin R n that are not necessarily Reinhardt domains. Corollary 1.5.
Let X Ω ⊂ R n and X Ω ⊂ R m be convex toric domains associated withbounded relative open subsets Ω ⊂ R n ≥ and Ω ⊂ R m ≥ , respectively. Then X Ω × X Ω is equalto the convex toric domain X Ω × Ω , and for any normalized symplectic capacity c on R n +2 m there holds c ( X Ω × X Ω ) = min { c ( X Ω ) , c ( X Ω ) } . The same conclusion holds true after Ω and Ω are replaced by Ω and Ω , respectively. This is a direct consequence of [3, (3.8)] and Theorem 1.4. In Section 2 we shall prove itwith only Theorem 1.4.For each p ∈ [1 , ∞ ] let k · k p denote the l p -norm in R n defined by k x k p := n X i =1 | x i | p ! /p if p < ∞ , k x k ∞ := max i | x i | . Then the open unit ball B np = { x = ( x , · · · , x n ) ∈ R n | k x k p < } is a convex balanced regionin R n . It was proved in [24, Theorem 7] that for a balanced region A ⊂ R n there exists asymplectomorphism between X | A | and the Lagrangian product B n ∞ × L A defined by B n ∞ × L A = (cid:8) ( x , · · · , x n , y , · · · , y n ) ∈ R n | ( x , · · · , x n ) ∈ B n ∞ , ( y , · · · , y n ) ∈ A (cid:9) , where 4 | A | = { (4 x , · · · , x n ) | ( x , · · · , x n ) ∈ | A |} . By this and Theorem 1.4 (resp. Corol-lary 1.5) we may, respectively, obtain two claims of the following Corollary 1.6.
For a convex balanced region A ⊂ R n and any normalized symplectic capacity c on R n there holds c ( B n ∞ × L A ) = 4 min {k e i k ∗| A | | i = 1 , · · · , n } . In particular, c ( B np × L B n ∞ ) = c ( B n ∞ × L B np ) = 4 for every p ∈ [1 , ∞ ] (since the symplectomor-phism R n ∋ ( x, y ) ( − y, x ) ∈ R n maps B n ∞ × L B np onto B n ∞ × L B np ). Moreover, for convexbalanced regions A i ⊂ R n i , i = 1 , · · · , k , it holds that c (( B n ∞ × · · · × B n k ∞ ) × L ( A × · · · × A k )) = min i c ( B n i ∞ × L A i ) . Consequently, the convex domain ( B n ∞ × · · · × B n k ∞ ) × L ( A × · · · × A k ) satisfies Conjecture 1.1and so Conjecture 1.2 by the first claim. Clearly, this result is a partial generalization of [2, Theorem 5.2] since B n ∞ is equal to ✷ n therein. Note that convex subsets B n ∞ × L B np (1 ≤ p < ∞ ) are not Reinhardt domains in R n .Since B n is a convex balanced region in R n and is equal to ( B n ∞ ) ◦ , Corollary 1.6 impliesthe known equality case in Mahler’s conjecture, which can also be proved by a straightforwardcomputation because Vol( B n ) = 2 n /n ! and Vol( B n ∞ ) = 2 n by (4.15). This and Corollary 1.6suggest the following questions for each p ∈ (1 , ∞ ): Is Conjecture 1.2 for the convex domain B np × ( B np ) ◦ ⊂ R n true? Does Conjecture 1.3 for the ball B np hold true?They are affirmative as examples of the following Theorems 1.8, 1.7, respectively. Theorem 1.7 (Saint-Raymond [27]) . Suppose that a centrally symmetric convex domain K ⊂ R n is -unconditional. Then Vol( K × K ◦ ) > n n ! and equality holds if K is a Hanner polytope. ecall that in [27, 26, 30] a centrally symmetric convex domain K ⊂ R n is called 1 -unconditional if there exists a basis { η , · · · , η n } of R n such that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 a i η i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) K = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 ε i a i η i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) K for all scalars a i ∈ R and signs ε i ∈ {− , } , 1 i n , where k · k K is the norm on R n determined by K , that is, k x k K = min { t > | x ∈ tK } , x ∈ R n .For 1 ≤ p ≤ ∞ , the p -product of two centrally symmetric convex domains K ⊂ R n and M ⊂ R m is defined by K × p M := [ ≤ t ≤ (cid:16) (1 − t ) p K × t p M (cid:17) , which is also centrally symmetric and has the corresponding norm k ( x, y ) k K × p M = ( k x k pK + k y k pM ) p , ( x, y ) ∈ R n × R m . From this it is not hard to derive that the operator × p is associative. Moreover, if both K and M are 1-unconditional, so is K × p M . Note also that K × ∞ M = K × M and K × M = conv { ( K × { } ) ∪ ( { } × M ) } . The 1-product is also called free sum .A centrally symmetric convex domain K ⊂ R n is called a Hanner polytope if it is obtained bysuccessively applying Cartesian products and free sums to centered line segments in arbitraryorder. Hence every Hanner polytope in R n is an affine image of I × p · · · × p n − I , where I = [ − ,
1] and p i ∈ { , ∞} , i = 1 , · · · , n − Theorem 1.8.
Suppose that A ⊂ R n is -unconditional convex domain. Then A × L A ◦ satisfies Conjecture 1.2, precisely, c ( A × L A ◦ ) ( n !Vol( A × L A ◦ )) n (1.7) for any normalized symplectic capacity c on R n . Recall that an ellipsoid in an n -dimensional normed space E is defined as a subset Q ⊂ E which is the image of B n by a line isomorphism (cf. [22, page 27]). We call the image of B np by a linear isomorphism of R n a l p -ellipsoid with p ∈ [1 , ∞ ]. Corollary 1.9.
For a l p -ellipsoid Q = Υ( B np ) ⊂ R n there holds c ( Q × L Q ◦ ) = 4 ≤ ( n !Vol( Q × L Q ◦ )) n (1.8) for any normalized symplectic capacity c on R n . In particular, Conjecture 1.2 holds for theconvex domain D = Q × Q ◦ . Since the Mahler volume is affine invariant, 4 ≤ ( n !Vol( Q × L Q ◦ )) n if and only if 4 ( n !Vol( B np × L ( B np ) ◦ )) n . The latter follows from (1.7). In Section 4 we shall give a directcalculus proof of the inequality. Organization of the paper . In Section 2 we prove Theorem 1.4 and Corollary 1.5. Next,we give proofs of Theorem 1.8 and Corollary 1.9 in Section 3. A direct proof of the Mahlerconjecture for l p -balls is given in Section 4. Finally, Section 5 includes some concluding re-marks. cknowledgments . We would like to thank Dr. Matthias Schymura for telling us that theMahler conjecture for l p -balls is true as an example of a result by J. Saint-Raymond [27], andfor sending us his beautiful Diplomarbeit, which is very helpful to us because researches onthe Mahler conjecture before March 2008 were explained explicitly. Proof of Theorem 1.4.
By [8, Theorem 1.6] and [9, Theorem 3.1], it holds that w G ( X Ω ) = min ( k v k ∗ Ω (cid:12)(cid:12)(cid:12) v = ( v , · · · , v n ) ∈ Z n ≥ , n X i =1 v i = 1 ) . (2.1)Let c be an arbitrarily given normalized symplectic capacity on R n . Then c ( X Ω ) > w G ( X Ω )by the normalization axiom of the symplectic capacity. Next let us show that c ( X Ω ) min ( k v k ∗ Ω (cid:12)(cid:12)(cid:12) v = ( v , · · · , v n ) ∈ Z n ≥ , n X i =1 v i = 1 ) . (2.2)Let { e i } ni =1 be the standard basis in R n , where e i = (0 , · · · , , , , · · · ,
0) with only the i -thcomponent non-zero, and equal to 1, i = 1 , · · · , n . Write L i = k e i k ∗ Ω and defineΩ ⋆i = { x ∈ R n ≥ | h e i , x i L i } , i = 1 , · · · , n. Then for each i , Ω ⊂ Ω ⋆i by the definition of k e i k ∗ Ω , and there exists an obvious symplectomor-phism from X Ω ⋆i = { ( z , · · · , z n ) ∈ C n = R n | π | z i | L i } onto Z n ( p L i /π ). It follows fromthe monotonicity and conformality of symplectic capacities that c ( X Ω ) ≤ c ( X Ω ⋆i ) = c ( Z n ( p L i /π )) = L i π c ( Z n (1)) = L i , i = 1 , · · · , n and so c ( X Ω ) ≤ min i L i . Note that each vector v = ( v , · · · , v n ) ∈ Z n ≥ with P ni =1 v i = 1 musthave form e j for some j ∈ { , · · · , n } . We get (2.2) and therefore c ( X Ω ) = min ( k v k ∗ Ω (cid:12)(cid:12)(cid:12) v = ( v , · · · , v n ) ∈ Z n ≥ , n X i =1 v i = 1 ) = min {k e i k ∗ Ω | i = 1 , · · · , n } (2.3)since k v k ∗ Ω = k v k ∗ Ω .Finally, we also need to prove c ( X Ω ) = c ( X Ω ). Clearly, c ( X Ω ) ≤ c ( X Ω ) by the monotonicityof symplectic capacities. Since X Ω is open and has the closure X Ω it follows from the definitionof the Gromov width w G in (1.1) that w G ( X Ω ) = w G ( X Ω ). This, and (2.1) and (2.3) yield c ( X Ω ) = w G ( X Ω ) = w G ( X Ω ) ≤ c ( X Ω )and hence c ( X Ω ) = c ( X Ω ). Now the proof is complete. Remark 2.1.
Let X Ω be a concave toric domain associated to a relative open subset Ω ⊂ R n ≥ .By [8, Theorem 1.14 & Corollary 1.16] and [9, Theorem 3.1], we have w G ( X Ω ) = c CH1 ( X Ω )= max ( [ v ] Ω (cid:12)(cid:12)(cid:12) v = ( v , · · · , v n ) ∈ Z n> , n X i =1 v i = n ) = inf ( n X i =1 w i | w = ( w , · · · , w n ) ∈ ∂ Ω ∩ R n> ) = max { πr | B n ( r ) ⊂ X Ω } = w G ( X Ω ) . (2.4) or any normalized symplectic capacity c on R n , repeating the proof of Theorem 1.4 we get c ( X Ω ) ≤ min ( k v k ∗ Ω (cid:12)(cid:12)(cid:12) v = ( v , · · · , v n ) ∈ Z n ≥ , n X i =1 v i = 1 ) = min {k e i k ∗ Ω | i = 1 , · · · , n } . Clearly, we have also c ( X Ω ) ≤ c ( X conv(Ω) ) and c ( X conv(Ω) ) ≤ min {k e i k ∗ conv(Ω) | i = 1 , · · · , n } = min {k e i k ∗ Ω | i = 1 , · · · , n } . This final equality easily follows from (1.5).If A ⊂ R n is a concave balanced region, since the Lagrangian product B n ∞ × L A is sym-plectomorphic to X | A | ([24, Theorem 7]), from (2.4) we get w G ( B n ∞ × L A ) = c CH1 ( B n ∞ × L A ) = 4 inf ( n X i =1 w i | w = ( w , · · · , w n ) ∈ ( ∂ | A | ) ∩ R n> ) . Proof of Corollary 1.5.
Since we can write X Ω = { ( z n +1 , · · · , z m + n ) ∈ C m | ( π | z n +1 | , · · · , π | z m + n | ) ∈ Ω } , then X Ω × X Ω = { ( z , · · · , z m + n ) ∈ C n + m | ( z , · · · , z n ) ∈ X Ω , ( z n +1 , · · · , z m + n ) ∈ X Ω } = X Ω × Ω and thus c ( X Ω × X Ω ) = c ( X Ω × Ω ). By Theorem 1.4, we get c ( X Ω × Ω ) = min {k e i k ∗ Ω × Ω | i = 1 , · · · , n + m } , where { e i } n + mi =1 is the standard orthogonal basis of R n + m . But for i = 1 , · · · , n , k e i k ∗ Ω × Ω = sup {h e i , x i | x = ( x , · · · , x n + m ) ∈ Ω × Ω } = sup { x i | x = ( x , · · · , x n + m ) ∈ Ω × Ω } = sup { x i | x = ( x , · · · , x n ) ∈ Ω } = k e i k ∗ Ω . Hence we arrive atmin {k e i k ∗ Ω × Ω | i = 1 , · · · , n } = min {k e i k ∗ Ω | i = 1 , · · · , n } = c ( X Ω ) . Similarly, we have min {k e i k ∗ Ω × Ω | i = n + 1 , · · · , n + m } = c ( X Ω ). Therefore c ( X Ω × Ω ) = min { c ( X Ω ) , c ( X Ω ) } . This and Theorem 1.4 also lead to the second conclusion.
Proof of Theorem 1.8.
We begin with the following lemma.
Lemma 3.1.
For a convex balanced region A ⊂ R n and any normalized symplectic capacity c on R n , there holds c ( A × L A ◦ ) . roof. Let r = max {k e i k ∗| A | | i = 1 , · · · , n } . By (1.5) we deduce that | A | ⊂ [0 , r ] n . This andthe definition of the balanced region imply that A ⊂ rB n ∞ . It follows from the monotonicityand conformality of symplectic capacities that c ( A × L A ◦ ) c (( rB n ∞ ) × L A ◦ ) = r c ( B n ∞ × L ( 1 r A ◦ )) . (3.5)Next, we claim that A ◦ is also a convex balanced region. It suffices to prove that A ◦ is abalanced region. In fact, for any ( y , · · · , y n ) ∈ A ◦ , since A is symmetric with respect to allcoordinate hyperplanes, we have { y , − y } × { y , − y } × · · · × { y n , − y n } ∈ A ◦ . (3.6)Moreover, for any y, y ′ ∈ A ◦ , we derive h ty + (1 − t ) y ′ , x i = t h y, x i + (1 − t ) h y ′ , x i , ∀ x ∈ A, ∀ < t < , that is, A ◦ is convex set. From this and (3.6) we derive[ −| y | , | y | ] × [ −| y | , | y | ] × · · · × [ −| y n | , | y n | ] ∈ A ◦ , namely, A ◦ is a balanced region.Now from Corollary 1.6 and (3.5) we deduce c ( A × L A ◦ ) ≤ r c ( B n ∞ × L ( 1 r A ◦ ))= 4 r min {k e i k ∗| A ◦ | | i = 1 , · · · , n } . (3.7)It remains to show that min {k e i k ∗| A ◦ | | i = 1 , · · · , n } r . Let r = k e j k ∗| A | for some 1 ≤ j ≤ n .Take a > ae j ∈ | A | . Then h ae j , x i ≤ ∀ x ∈ A ◦ . In particular, h e j , x i ≤ a ∀ x ∈| A ◦ | . This shows k e j k ∗| A ◦ | a . Note that k e j k ∗| A | > a > k e j k ∗| A | . We get k e j k ∗| A ◦ | k e j k ∗| A | = r , and thereforemin {k e i k ∗| A ◦ | | i = 1 , · · · , n } ≤ k e j k ∗| A ◦ | ≤ r . This and (3.7) lead to the desired result.By Theorem 1.7, if a centrally symmetric convex domain A ⊂ R n is a balanced region,in particular a Hanner polytope, then Vol( A × L A ◦ ) > n n ! and therefore A × L A ◦ satisfiesConjecture 1.2, i.e., c ( A × L A ◦ ) ( n !Vol( A × L A ◦ )) n (3.8)for any normalized symplectic capacity c on R n .Now assume that A ⊂ R n is 1-unconditional convex domain with basis { η , · · · , η n } . Let { e , · · · , e n } be the standard basis of R n , and let Υ ∈ GL( n, R ) map η i to e i for i = 1 , · · · , n .Since k x k Υ( A ) = k Υ − x k A for any x ∈ R n , a straightforward computation shows that Υ( A ) ⊂ R n is 1-unconditional convex domain with basis { e , · · · , e n } . It follows that k ( x , · · · , x n ) k Υ( A ) = k ( | x | , · · · , | x n | ) k Υ( A ) , ∀ x ∈ R n , which means that the convex domain Υ( A ) ⊂ R n is a balanced region. By (3.8) we get c (Υ( A ) × L (Υ( A )) ◦ ) ( n !Vol(Υ( A ) × L (Υ( A )) ◦ )) n (3.9) or any normalized symplectic capacity c on R n . Denote by Υ T the transpose of Υ ∈ GL( n, R )with respect to the inner product h· , ·i in R n . ThenΦ Υ : ( R n , ω ) → ( R n , ω ) , ( x, y ) (Υ x, (Υ T ) − y ) (3.10)is a symplectomorphism. By the definition of the polar it is easy to check that(Υ( A )) ◦ = { x ∈ R n | h y, x i ≤ ∀ y ∈ Υ( A ) } = { (Υ T ) − u | u ∈ A ◦ } = (Υ T ) − ( A ) ◦ . Then Υ( A ) × (Υ( A )) ◦ = Φ Υ ( A × A ◦ ), Vol((Υ( A )) ◦ ) = | det(Υ T ) − | Vol( A ◦ ) and soVol(Υ( A ) × (Υ( A )) ◦ ) = Vol(Υ( A ))Vol((Υ( A )) ◦ ) = Vol( A )Vol( A ◦ ) = Vol( A × L A ◦ ) . From these and (3.9) we derive (1.7). Theorem 1.8 is proved.
Proof of Corollary 1.9.
Since every closed l p -ball B np is a 1-unconditional convex domain withbasis { e i } ni =1 in R n , for any normalized symplectic capacity c on R n we derive from (1.7) that c ( B np × L ( B np ) ◦ ) ( n !Vol( B np × L ( B np ) ◦ )) n (3.11)and therefore c ( A × L A ◦ ) ( n !Vol( A × L A ◦ )) n . (3.12)If p = 1 or ∞ , Corollary 1.6 has yielded c ( B np × L ( B np ) ◦ ) = 4. For 1 < p < ∞ , we have w G ( B np × L ( B np ) ◦ ) ≥ c ( A × L A ◦ ) ≥ w G ( A × L A ◦ ) = w G ( B np × L ( B np ) ◦ ) ≥ ∀ p ∈ [1 , ∞ ] . This and the first inequality in (3.12) lead to equality in (1.8). l p -balls In this section we shall prove the following.
Theorem 4.1.
Let Q = Υ( B np ) ⊂ R n be a l p -ellipsoid with Υ ∈ GL( n, R ) . If n = 1 then Vol( Q × Q ◦ ) = Vol( Q )Vol( Q ◦ ) ≡ for all p ∈ [1 , ∞ ] . If n ≥ then there holds Vol( Q × Q ◦ ) = Vol( Q )Vol( Q ◦ ) ≥ n n ! (4.13) for all p ∈ [1 , ∞ ] , and the equality holds if and only if p = 1 or p = ∞ . As the arguments below (3.10) we only need to prove the case Υ = id R n , that is: Claim 4.2.
For n = 1 , Vol( B np × ( B np ) ◦ ) = Vol( B n × ( B n ) ◦ ) = 4 ∀ p ∈ [1 , ∞ ] . If n ≥ then Vol( B np × ( B np ) ◦ ) = Vol( B np )Vol(( B np ) ◦ ) ≥ n /n ! , ∀ p ∈ [1 , ∞ ] , (4.14) and the equality in (4.14) holds if and only if p = 1 or p = ∞ . This is a special example of Theorem 1.7 because B np is a centrally symmetric convexdomain R n with 1-unconditional basis { e i } ni =1 . However, we here give a simple calculus proofof it.Since ( B np ) ◦ = B nq with q = p/ ( p − , ∋ p q = p/ ( p − ∈ [2 , ∞ ] is ahomeomorphism, by symmetry it suffices to prove Claim 4.2 for p ∈ [1 , y [22, (1.17)], we haveVol( B np ) = (cid:18) (cid:18) p (cid:19)(cid:19) n (cid:18) Γ (cid:18) np (cid:19)(cid:19) − (4.15)and so Vol(( B np ) ◦ ) = (cid:18) (cid:18) p/ ( p − (cid:19)(cid:19) n (cid:18) Γ (cid:18) np/ ( p − (cid:19)(cid:19) − = (cid:18) (cid:18) − p (cid:19)(cid:19) n (cid:18) Γ (cid:18) n + 1 − np (cid:19)(cid:19) − and Vol( B np )Vol(( B np ) ◦ ) = 4 n (cid:16) Γ (cid:16) p (cid:17)(cid:17) n (cid:16) Γ (cid:16) − p (cid:17)(cid:17) n Γ (cid:16) np (cid:17) Γ (cid:16) n + 1 − np (cid:17) . Taking the derivative of the function [1 , ∋ p Vol( B np )Vol(( B np ) ◦ ) we get ddp Vol( B np )Vol(( B np ) ◦ )= 4 n np Γ(1 + 1 p ) n − Γ(2 − p ) n − [Γ(1 + p )Γ ′ (2 − p ) − Γ ′ (1 + p )Γ(2 − p )]Γ(1 + np )Γ( n + 1 − np )+4 n np Γ(1 + 1 p ) n Γ(2 − p ) n [Γ ′ (1 + np )Γ( n + 1 − np ) − Γ ′ ( n + 1 − np )Γ(1 + np )]Γ(1 + np ) Γ( n + 1 − np ) . Recall that the formula Γ ′ ( x ) = Γ( x ) ψ ( x ) ∀ x >
0, where ψ -function is defined by ψ ( x ) = lim n →∞ ( ln n − n X k =0 x + k ) = Z ∞ [ e − t − (1 + t ) − x ] t − dt (Gauss intergral formula)= − γ + Z − t x − − t dt (Dirichlet formula)where γ is Euler constant. We can immediately deduce ddp Vol( B np )Vol(( B np ) ◦ )= 4 n np Γ(1 + 1 p ) n Γ(2 − p ) n ψ (2 − p ) − ψ (1 + p ) + ψ (1 + np ) − ψ ( n + 1 − np )Γ(1 + np )Γ( n + 1 − np )= np (cid:20) ψ (2 − p ) − ψ (1 + 1 p ) + ψ (1 + np ) − ψ ( n + 1 − np ) (cid:21) Vol( B np )Vol(( B np ) ◦ ) . Denote by Φ n ( p ) the function in the square brackets. Then Φ ( p ) ≡ Claim 4.3.
When n ≥ , Φ n (2) = 0 and Φ n ( p ) > for any ≤ p < . We first admit this. Then the function [1 , ∋ p Vol( B np )Vol(( B np ) ◦ ) is strictly monotonouslyincreasing for each integer n ≥
2. Moreover, Vol( B n ) = 2 n /n ! and Vol( B n ∞ ) = 2 n . Claim 4.3immediately leads to the second conclusion in Claim 4.2. roof of Claim 4.3. Since ψ ( x + 1) = ψ ( x ) + x , then Φ n (2) = 0 and Φ n (1) = P nk =2 1 k . Wealways assume 1 < p < ψ ( s + 1) = − γ + Z − x s − x dx. It follows that ψ (2 − p ) − ψ (1 + 1 p ) = Z x /p − x − /p − x dx, (4.16) ψ (1 + np ) − ψ ( n + 1 − np ) = Z x n − n/p − x n/p − x dx = Z y − /p − y /p − y − y − y /n n y n − dy (4.17)by setting x n = y . For convenience let a = 1 /n and f ( y ) := 1 − y − y /n n y n − = a − y − y a y a − . A straightforward computation leads to f ′ ( y ) = a (cid:18) − y − y a (cid:19) ′ y a − + a − y − y a ( a − y a − = ay a − − (1 − y a ) − (1 − y )( − ay a − )(1 − y a ) + a − y − y a ( a − y a − = ay a − (1 − y a ) (cid:0) − (1 − y a ) + ay a − (1 − y ) + (1 − y )( a − y − (1 − y a ) (cid:1) = ay a − (1 − y a ) (cid:0) ( a − y − −
1) + y a − − (cid:1) = ay a − (1 − y a ) (cid:18) y ( a − ay − y a ) (cid:19) . Let g ( y ) = a − ay − y a . Then g (0) = a − < g (1) = 0 and g ′ ( y ) = − a + ay a − > < y <
1. It follows that g ( y ) < f ′ ( y ) < < y < y → f ( y ) = 1. Hence f ( y ) > < y <
1. Using (4.16) and (4.17) we deduce thatΦ n ( p ) = ψ (2 − p ) − ψ (1 + 1 p ) + ψ (1 + np ) − ψ ( n + 1 − np )= Z y /p − y − /p − y dy + Z y − /p − y /p − y f ( y ) dy = Z y − /p − y /p − y ( f ( y ) − dy > y − /p − y /p = y /p ( y − /p − < < y < < p <
2. Claim 4.3 isproved.
Remark 5.1.
For 1 p < ∞ , X p = { ( x, y ) ∈ R × R | k x k p + k y k p } is called the l p -sumof two Langrangian open unit discs B , where k · k denotes the standard Euclidean norm on . If p = ∞ , X ∞ = { ( x, y ) ∈ R × R | max {k x k , k y k} < } is exactly the Lagrangian product B × L B . For any normalized symplectic capacity c on R and p ∈ [1 , ∞ ] it easily followsfrom [21, 23, 8, 9] that c ( X p ) = π (1 / /p , p ∈ [1 , , p ) Γ(1 + p ) , p ∈ [2 , ∞ ) , , p = ∞ . (5.1)In particular, X p satisfies Conjecture 1.1.In fact, for p ∈ [1 , ∞ ), by [21, Theorem 5] X p is symplectomorphic to X Ω p , where Ω p is therelatively open set in R > bounded by the coordinate axes and the curve γ p parametrized by(2 πv + g p ( v ) , g p ( v )) , for v ∈ [0 , (1 / /p ] , ( g p ( − v ) , − πv + g p ( − v )) , for v ∈ [ − (1 / /p , , where g p : [0 , (1 / /p ] → R is the function defined by g p ( v ) := 2 Z − √ − v p /p ( + √ − v p ) /p r (1 − r p ) /p − v r dr. For p = ∞ , Theorem 3 in [23] (with the notations in [21, Theorem 6]) claimed that X ∞ = B × L B is symplectictomorphic to X Ω ∞ , where Ω ∞ is the the relatively open set in R > bounded by the coordinate axes and the curve γ ∞ parametrized by2( p − v + v ( π − arccos v ) , p − v − v arccos v ) , for v ∈ [ − , α/ − α cos( α/ , α/
2) + (2 π − α ) cos( α/ α ∈ [0 , π ], see[23, Theorem 3]). Moreover, by [21, Proposition 8], we also know that the toric domain X Ω p isconvex for p ∈ [1 , p ∈ [2 , ∞ ]. Hence for any normalized symplectic capacity c on R , [9, Theorem 1.4] and [21, Theorem 1] lead to the first two cases in (5.1), and thethird case follows from [9, Theorem 1.4] and [8, Theorem 1.14], c ( X ∞ ) = max { [ v ] Ω ∞ | v ∈ Z > , X i v i = 2 } = inf (cid:8) w + w | w = ( w , w ) ∈ ∂ Ω ∞ ∩ R > (cid:9) = 4 . Remark 5.2.
Suppose that each of symplectic manifolds X (1) , · · · , X ( m ) is either a convextoric domain or 4-dimensional concave toric domain or equal to X p as in (5.1). Since eachconvex or concave toric domain or X p is star-shaped, then for any normalized symplecticcapacity c on R n with 2 n = P mi =1 dim X ( i ) , from [3, (3.8)], Theorem 1.4 and (5.1) we derive c EH1 ( m Y i =1 X ( i ) ) = min i c ( X ( i ) ) . Remark 5.3.
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