Continuous quantitative Helly-type results
aa r X i v : . [ m a t h . M G ] A ug CONTINUOUS QUANTITATIVE HELLY-TYPE RESULTS
TOM ´AS FERNANDEZ VIDAL, DANIEL GALICER, AND MARIANO MERZBACHER
Abstract.
Brazitikos’ results on quantititative Helly-type theorems (for thevolume and for the diameter) rely on the work of Srivastava on approxi-mate John’s decompositions with few vectors. We change this technique bya stronger recent result due to Friedland and Youssef that allow us to obtainHelly-type versions which are sensitive to the number of convex sets involved. Introduction
Helly’s classical theorem states that if C = { C i : i ∈ I } is a finite family of at least n + 1 convex sets in R n and if any n + 1 members of C have non-empty intersectionthen T i ∈ I C i is non-empty. In general, a Helly-type property is a property Π forwhich there exists a number s ∈ N such that if { C i : i ∈ I } is a finite family ofcertain objects and every subfamily of s elements fulfills Π, then the whole familyfulfills Π.In the eighties, B´ar´any, Katchalski and Pach proved the following quantitative“volume version” of Helly’s theorem [BKP82, BKP84]: Let C = { C i : i ∈ I } be a finite family of convex sets in R n . If the intersectionof any n or fewer members of H has volume greater than or equal to , then vol( T i ∈ I C i ) ≥ c ( n ) , where c ( n ) > is a constant depending only on n . Thus, the previous result express the fact that “the intersection has large volume” is a Helly-type property for the family of convex sets.Since every (closed) convex set is the intersection of a family of closed half-spaces; a simple compactness argument (see [BKP82]) shows that one can removethe restriction that C is finite and also assume that each convex set is a closedhalf-space i.e., { x ∈ R n : h x, v i i ≤ } , for some vector v i ∈ R n . Therefore, the theorem of B´ar´any et. al. is equivalent tothe following statement: Let H = { H i : i ∈ I } be a family of closed half-spaces in R n such that vol( T i ∈ I H i ) = 1 . There exist s ≤ n and i , . . . , i s ∈ I such that vol( H i ∩ · · · ∩ H i s ) /n ≤ c ( n ) , where c ( n ) > is a constant depending only on n . Mathematics Subject Classification.
Key words and phrases.
Helly-type results, convex bodies, approximate John’s decomposition. T i ∈ I H i H i ∩ · · · ∩ H i s Figure 1.
A convex body defined as the intersection of half-spaceswhich is enclosed by a convex set given by the intersection of a fewof them.Of course one cannot replace 2 n by 2 n − − / , / n in R n can be written as the intersection of the 2 n closedhalf-spaces H ± j := (cid:26) x : h x, ± e j i ≤ (cid:27) and that the intersection of any 2 n − c ( n ) ≤ n n for the constant c ( n ) andconjectured that one might actually have polynomial growth i.e., c ( n ) ≤ n d for anabsolute constant d >
0. Nasz´odi [Nas16] has verified this conjecture; namely, heproved that c ( n ) ≤ cn , where c > instead of 2.Moreover, Brazitikos showed in [Bra17a, Theorem 1.4.] that if we relax thecondition on the number s of half-spaces that we use (but still require that it isproportional to the dimension n ) one can improve significantly the estimate, givinga bound of order n . Theorem 1.1. [Bra17a, Theorem 1.4.]
There exists an absolute constant α > with the following property: for every family H = { H i : i ∈ I } of closed half-spacesin R n , H i = { x ∈ R n : h x, v i i ≤ } , with vol( T H i ∈ I H i ) = 1 , there exist s ≤ αn and i , . . . , i s ∈ I such that vol( H i ∩ · · · ∩ H i s ) /n ≤ cn, where c > is an absolute constant. B´ar´any, Katchalski and Pach also studied the question whether “ the intersectionhas large diameter ” is a sort of Helly-type property for convex sets. They providedthe following quantitative answer to this question:
ONTINUOS QUANTITATIVE HELLY-TYPE RESULTS 3
Let { C i : i ∈ I } be a family of closed convex sets in R n such that diam (cid:0)T i ∈ I C i (cid:1) =1 . There exist s n and i , . . . , i s ∈ I such that diam ( C i ∩ · · · ∩ C i s ) ( cn ) n/ , where c > is an absolute constant. In the same work the authors conjectured that the bound ( cn ) n/ should bepolynomial in n . Leaving aside the requirement that s n , Brazitikos in [Bra17b]provided the following relaxed positive answer: Theorem 1.2.
There exists an absolute constant α > with the following property:if { C i : i ∈ I } is a finite family of convex bodies in R n with diam (cid:0)T i ∈ I C i (cid:1) = 1 ,there exist s αn and i , . . . i s ∈ I such that diam( C i ∩ · · · ∩ C i s ) cn / , where c > is an absolute constant. It should be mentioned that when symmetry is assumed better bounds in bothproblems can be obtained.Brazitikos’ proofs of Theorem 1.1 and Theorem 1.2 rely on the work of Batson,Spielman and Srivastava on approximate John’s decompositions with few vectors[BSS12]. For Theorem 1.1, this is successfully combined with a new and very usefulestimate for corresponding ‘approximate’ Brascamp-Lieb-type inequality while, forTheorem 1.2, the argument is based on a clever lemma of Barvinok from [Bar14].This lemma in turn, exploits again the theorem of Batson et. al. or to be precise,a more delicate version of Srivastava from [Sri12].Of course if one is willing to further relax the number of convex sets involved inthe statements of Theorems 1.1 and 1.2, then one should expect to obtain betterbounds/estimates. The aim of this note is to present the following continuous quantitative Helly-type results (i.e., Helly-type results which are sensitive to thenumber of sets considered).
Theorem 1.3. (Continuous Helly-type theorem for the volume) Let ≤ δ ≤ ,there is an absolute constant α > with the following property: for every n ∈ N and every family H = { H i : i ∈ I } of closed half-spaces in R n , H i = { x ∈ R n : h x, v i i ≤ } , with vol( T i ∈ I H i ) = 1 , there exists s ≤ αn δ and i , . . . , i s ∈ I such that vol( H i ∩ · · · ∩ H i s ) /n ≤ d n n − δ , where d n → as n → ∞ . Theorem 1.4. (Continuous Helly-type theorem for the diameter) Let ≤ δ ≤ ,there is an absolute constant α > with the following property: for every n ∈ N and every finite family { C i : i ∈ I } of convex bodies in R n with diam (cid:0)T i ∈ I C i (cid:1) = 1 ,there exist s αn δ and i , . . . i s ∈ I such that diam( C i ∩ · · · ∩ C i s ) cn − δ , where c > are absolute constant. TOM´AS FERNANDEZ VIDAL, DANIEL GALICER, AND MARIANO MERZBACHER
Note that in both theorems we recover the previous mentioned results whenthe number of sets is linear in n (i.e., when δ = 1). If the number of sets is n then the bounds are the known ones which, of course, follow by directly applyingJohn’s classical theorem. Therefore, the dependencies in the exponent of bothresults obtained seem to be accurate. Moreover, for a linear number of spaces (i.e., δ = 1) the constant that appears in Theorem 1.3 is better than the one in [Bra17a,Theorem 1.4.], since d n → n goes to infinity.To obtain Theorems 1.3 and 1.4 we carefully follow Brazitikos’s proofs of Theo-rems 1.1 and 1.2 but instead of using Batson et. al. or Srivastava’s statment on theapproximate John’s decomposition we replace it with the following stronger resultdue to Friedland and Youssef (who exploited the recent solution of the Kadison-Singer problem [MSS15], by showing that any n × m matrix A can be approximatedin operator norm by a submatrix with a number of columns of order the stable rankof A ). Theorem 1.5. [FY19, Theorem 4.1]
Let { u j , a j } ≤ j ≤ m be a John’s decomposi-tion of the identity i.e, the identity operator I n is decomposed in the form I n = P mj =1 a j u j ⊗ u j . Then for any ε > there exists a multi-set σ ⊂ [ m ] (i.e., it allowsrepetitions of the elements) with | σ | ≤ n/cε so that (1 − ε ) I n (cid:22) n | σ | X j ∈ σ ( u j − u ) ⊗ ( u j − u ) (cid:22) (1 + ε ) I n where u = | σ | P i ∈ σ u j satisfies k u k ≤ ε √ n , and c > is an absolute constant. In the words of Friedland and Youssef, Thorem 1.5 improves Srivastavas theorem[Sri12, Theorem 5] in three different ways. First the approximation ratio (1+ ε ) / (1 − ε ) can be made arbitrary close to 1 (while in Srivastavas result one could only geta (4 + ε )-approximation). Secondly, it gives an explicit expression of the weightsappearing in the approximation. Finally, there is a big difference in the dependenceon ε in the estimate of the norm of u : Srivastava obtains a similar bound but with ε replaced by √ ε . This behaviour on ε will be crucial for our purposes allowingus to obtain the bounds on our main results. With this at hand, we take the ε parameter small but depending explicitly on n .2. Notation and background
We refer to the book of Artstein-Avidan, Giannopoulos and V. Milman [AAGM15]for basic facts from convexity and asymptotic geometry.Recall that a convex body in R n is a compact convex subset K of R n withnon-empty interior. We say that the body K is symmetric if x ∈ K implies that − x ∈ K . For any set X we write conv( X ) for its convex hull. For convex body K we write p K for the Minkowski’s functional of K , that is p K ( x ) := inf { λ > x ∈ λK } . If 0 ∈ int( K ) then the polar body K ◦ of K is given by K ◦ := { y ∈ R n : h x, y i x ∈ K } . ONTINUOS QUANTITATIVE HELLY-TYPE RESULTS 5
Volume is denoted by vol( · ) and diameter by diam( · ) . We consider in R n the Eu-clidean structure h· , ·i and denote by k · k the corresponding Euclidean norm. Wewrite B n and S n − for the corresponding Euclidean unit ball and unit sphere re-spectively.We say that a convex body K is in John’s position if the ellipsoid of maximalvolume inscribed in K is the Euclidean unit ball B n . John’s classical theorem statesthat K is in John’s position if and only if B n ⊆ K and there exist u , . . . , u m ∈ bd( K ) ∩ S n − (contact points of K and B n ) and positive real numbers a , . . . , a m such that m X j =1 a j u j = 0and the identity operator I n is decomposed in the form(1) I n = m X j =1 a j u j ⊗ u j , where the rank-one operator u j ⊗ u j is simply ( u j ⊗ u j )( y ) = h u j , y i u j .If u , . . . , u m are unit vectors that satisfy John’s decomposition (1) with somepositive weights a j . Then, one has the useful equalities m X j =1 a j = tr( I n ) = n and m X j =1 a j h u j , z i = 1for all z ∈ S n − . Moreover,(2) conv { v , . . . , v m } ⊇ n B n . The body K is in L¨owner position if the minimal volume ellipsoid that containsit is the Euclidean ball B n . In that case, we also have a decomposition of theidentity as before.Given two matrices A, B ∈ R n × n we write A (cid:22) B whenever B − A is positivesemidefinte.The letters c, c ′ , C, C ′ etc. will always denote absolute positive constants whichmay change from line to line.3. Continuous Helly-type result for the volume: Theorem 1.3.
As mentioned above we follow the proof of [Bra17a, Theorem 1.4.]. We includeall the steps for completeness.The following Brascamp-Lieb type inequality for approximate John’s decompo-sition of the identity will be crucial.
Theorem 3.1. [Bra17a, Theorem 5.4]
Let γ > . Let u , · · · , u s ∈ S n − and a , · · · , a s > satisfy Id n (cid:22) A := s X j =1 a j u j ⊗ u j (cid:22) γId n TOM´AS FERNANDEZ VIDAL, DANIEL GALICER, AND MARIANO MERZBACHER and let k j = a j h A − u j , u j i > , ≤ j ≤ s . If f , · · · , f s : R −→ R + integrablefunctions then Z R n s Y j =1 f k j j ( h x, u j i ) d x ≤ γ n s Y j =1 Z R f j ( t ) d t k j . We now prove Theorem 1.3.
Proof. (of Theorem 1.3)Without loss of generality we assume that P := T i ∈ I H i is in John’s position.Therefore there exist J ⊆ I and vectors ( u j ) j ∈ J which are contact points between P and S n − and ( a j ) j ∈ J positive numbers, such that Id n = X j ∈ J a j u j ⊗ u j and X j ∈ J a j u j = 0 . Using Friedland and Youssef’s approximate decomposition, Theorem 1.5, we canfind a multi-set σ ⊆ J with | σ | ≤ ncε and a vector u = − | σ | P j ∈ σ u j such that(1 − ε ) Id n (cid:22) n | σ | X j ∈ σ ( u j + u ) ⊗ ( u j + u ) (cid:22) (1 + ε ) Id n , also satisfying that n | σ | P j ∈ σ u j + u = 0 and | u | ≤ ε √ n .We consider the vector w := u √ nε . Recall that n B n ⊆ conv { u j , j ∈ J } , thus k w k ≤ n and hence w ∈ conv { u j , j ∈ J } . By Carathodory’s Theorem, we knowthat there is τ ⊆ J , with | τ | ≤ n + 1 and ρ i > , i ∈ τ such that w = X i ∈ τ ρ i u i and X i ∈ τ ρ i = 1 . Also notice that, since u = − | σ | P j ∈ σ u j and P j ∈ σ | σ | = 1, − u ∈ conv { u j , j ∈ σ } . Therefore, we have that the segment h − u, u √ nε i is contained in conv { u j , j ∈ σ ∪ τ } . For j ∈ σ we define v j := r nn + 1 (cid:18) − u j , √ n (cid:19) and b j = n + 1 | σ | . Set v := − q nn +1 ( u, X j ∈ σ b j ( v j + v ) ⊗ ( v j + v ) = X j ∈ σ n | σ | (cid:18) − ( u j + u ) , √ n (cid:19) ⊗ (cid:18) − ( u j + u ) , √ n (cid:19) = P j ∈ σ n | σ | ( u j + u ) ⊗ ( u j + u ) √ n | σ | P j ∈ σ ( u j + u ) √ n | σ | P j ∈ σ ( u j + u ) t n | σ | P j ∈ σ n ! = (cid:18)P j ∈ σ n | σ | ( u j + u ) ⊗ ( u j + u ) 00 1 (cid:19) , which implies(3) (1 − ε ) Id n +1 (cid:22) X j ∈ σ b j ( v j + v ) ⊗ ( v j + v ) (cid:22) (1 + ε ) Id n +1 . ONTINUOS QUANTITATIVE HELLY-TYPE RESULTS 7
The sum P j ∈ σ b j ( v j + v ) ⊗ ( v j + v ) can be written as X j ∈ σ b j v j ⊗ v j + v ⊗ X j ∈ σ b j v j + X j ∈ σ b j v j ⊗ v + ( n + 1) v ⊗ v, and notice that since X j ∈ σ b j v j = X j ∈ σ n + 1 | σ | r nn + 1 (cid:18) − u j , √ n (cid:19) = r n + 1 n − X j ∈ σ n | σ | u j , | σ | X j ∈ σ √ n = r n + 1 n (cid:0) nu, √ n (cid:1) , we obtain that X j ∈ σ b j v j ⊗ v = r n + 1 n (cid:0) nu, √ n (cid:1) ⊗ r nn + 1 ( − u,
0) = (cid:18) − nu ⊗ u −√ nu t (cid:19) ,v ⊗ X j ∈ σ b j v j = (cid:18) − nu ⊗ u −√ nu (cid:19) , and ( n + 1) v ⊗ v = (cid:18) nu ⊗ u
00 0 (cid:19) .Hence, we can write Equation (3) as(1 − ε ) Id n +1 − T (cid:22) X j ∈ σ b j v j ⊗ v j (cid:22) (1 + ε ) Id n +1 , where T = v ⊗ (cid:16)P j ∈ σ b j v j (cid:17) + (cid:16)P j ∈ σ b j v j (cid:17) ⊗ v + ( n + 1) v ⊗ v = (cid:18) V zz (cid:19) , with V = − nu ⊗ u y z = −√ nu . Now, for ( x, t ) ∈ S n we have that h T ( x, t ) , ( x, t ) i = h ( V x + zt, h z, x i ) , ( x, t ) i = h ( V x, , ( x, t ) i + h ( zt, h z, x i ) , ( x, t ) i≤ h V x, x i + | ( zt, h z, x i ) || ( t, x ) | = h V x, x i + (cid:0) | zt | + h z, x i (cid:1) ≤ k V k| x | + (cid:0) | z | t + | z | | x | (cid:1) ≤ k V k + | z | (cid:0) t + | x | (cid:1) = k V k + | z || ( x, t ) | = k V k + | z | = n | u | + √ n | u |≤ n ε n + √ n ε √ n = 4 ε ε ≤ ε, for ε small enough (say ε ≤ ). So, k T k ≤ ε , and hence Equation (3) implies that(1 − ε ) Id n +1 (cid:22) A := X j ∈ σ b j v j ⊗ v j (cid:22) (1 + 2 ε ) Id n +1 , TOM´AS FERNANDEZ VIDAL, DANIEL GALICER, AND MARIANO MERZBACHER or equivalently Id n +1 (cid:22) X j ∈ σ b j − ε v j ⊗ v j (cid:22) γId n +1 , with γ = ε − ε . Applying Theorem 3.1, if f j : R → R + are measurable functions,then Z R n +1 Y j ∈ σ f k j j ( h x, v j i ) d x ≤ γ n +12 Y j ∈ σ Z R f j ( t ) d t k j , where k j = b j − ε *(cid:18) − ε A (cid:19) − v j , v j + = b j h A − v j , v j i . Since A − (cid:22) − ε Id n +1 , we have that k j b j ≤ − ε . Now for j ∈ σ we consider f j ( t ) := e − bjkj t χ [0 , ∞ ) ( t ). So, Z R n +1 Y j ∈ σ f k j j ( h x, v j i ) d x ≤ γ n +12 Y j ∈ σ Z R f j ( t ) d t k j = γ n +12 Y j ∈ σ k j b j k j ≤ γ n +12 − ε ) P j ∈ σ k j = γ n +12 − ε ) n +1 = (cid:18) ε (1 − ε ) (cid:19) n +12 Set Q = \ i ∈ σ ∪ τ H i = { x ∈ R n : h x, u j i < , j ∈ σ ∪ τ } , and let y = ( x, r ) ∈ R n +1 . Assume that r > x ∈ r √ n Q . Then we have that h x, u j i < r √ n for every j ∈ σ , which implies that h y, v j i > j ∈ σ , andthen Q j ∈ σ f k j j ( h y, v j i ) >
0. We also have that * | σ | X j ∈ σ u j , x + = h− u, x i = 2 √ nε h− w, x i = 2 √ nε * − X i ∈ τ ρ i u i , x + ≥ − √ nε X i ∈ τ ρ i ! r √ n = − εr . ONTINUOS QUANTITATIVE HELLY-TYPE RESULTS 9
Thus, if y = ( x, r ) ∈ r √ n Q × (0 , ∞ ), then Y j ∈ σ f k j j ( h y, v j i ) = exp − X j ∈ σ b j (cid:18) r √ n + 1 − r nn + 1 h x, u j i (cid:19) = exp − r √ n + 1 X j ∈ σ b j exp √ n √ n + 1 * x, | σ | X j ∈ σ u j + ≥ e − r √ n +1 e −√ n √ n +1 rε = e − r √ n +1 ( ε √ n ) . Now, by Theorem 3.1,vol( Q ) n n ∞ Z r n e − r √ n +1 ( ε √ n ) d r = ∞ Z Z r √ n Q e − r √ n +1 ( ε √ n ) d x d r ≤ Z R n +1 Y j ∈ σ f k j j ( h y, v j i ) d y ≤ (cid:18) ε (1 − ε ) (cid:19) n +12 . Using that B n ⊆ P , and the fact that ∞ Z r n e − r √ n +1 ( ε √ n ) d r = n !( n + 1) n +12 (cid:0) ε √ n (cid:1) n +1 , we obtain, by taking 1 + ε ′ = ε (1 − ε ) ,vol (cid:0) \ i ∈ σ ∪ τ H i (cid:1) = vol( Q ) ≤ (1 + ε ′ ) n +12 n n ( n + 1) n +12 (cid:0) ε √ n (cid:1) n +1 n ! vol( P )vol( B n )= (1 + ε ′ ) n +12 n n ( n + 1) n +12 (cid:0) ε √ n (cid:1) n +1 n ! Γ (cid:0) n + 1 (cid:1) vol( P ) π n . By Stirling’s formula we get, for a constant
C >
0, the inequalityvol( \ i ∈ σ ∪ τ H i ) ≤ C (1 + ε ′ ) n +12 n n ( n + 1) n +12 (cid:0) ε √ n (cid:1) n +1 π n √ πn (cid:0) ne (cid:1) n n r πn (cid:16) n e (cid:17) n vol( P )= C (1 + ε ′ ) n +12 (cid:0) ε √ n (cid:1) n +1 n n n ( n + 1) n +12 n n n (cid:16) e π (cid:17) n √ P )= C (1 + ε ′ ) n +12 (cid:18) ε √ n (cid:19) n +1 ( n + 1) n +12 (cid:16) e π (cid:17) n √ P ) . Fix ε := n (1 − δ ) / , using that 1 + ε ′ = ε (1 − ε ) we have(1 + ε ′ ) (cid:18) ε √ n (cid:19) e π = (1 + ε ′ ) (cid:18) n (2 − δ ) / (cid:19) e π < cn − δ . Therefore,vol( \ i ∈ σ ∪ τ H i ) ≤ Cn n (cid:18) n (cid:19) n √ n + 1 n (2 − δ ) / n n (2 − δ ) / √ π √ e vol( P ) ≤ C r e ( n + 1) πe n n n (2 − δ ) / n n (2 − δ ) / vol( P )= C √ n + 1 n (2 − δ ) / √ π | {z } C n n n (3 − δ ) / vol( P ) . We conclude that vol( \ i ∈ σ ∪ τ H i ) ≤ C n n n (3 − δ ) / vol( P ) , where the intersection is taken over at most | σ ∪ τ | ≤ ncε + n + 1 = n δ c + n + 1 ≤ αn δ half-spaces. Since the constant C n is of order n (3 − δ ) / , we have that d n := C /nn → n → ∞ .It should be mentioned that the case δ = 2 is of course easier (we just useJohn’s decomposition of the identity and the classical Brascamp-Lieb inequalitydirectly). (cid:3) Continuous Helly-type theorem for the diameter
To obtain Theorem 1.4 we prove the following proposition, which is a continuousversion of [Bra17b, Proposition 4.2.]. We feel it is interesting in its own right. Againwe include all the steps for completeness.
Proposition 4.1.
Let ≤ δ ≤ . If K is a convex body whose minimal volumeellipsoid is the Euclidean unit ball, then there is a subset X ⊆ K ∩ S n − of cardinality card ( X ) ≤ αn δ and K ⊆ B n ⊆ Cn − δ conv( X ) , where α, C > are absolute constant.Proof. By John’s theorem there exist v j ∈ K ∩ S n − and a j > j ∈ J , such that I n = X j ∈ J a j v j ⊗ v j and X j ∈ J a j v j = 0 . Let ε > σ ⊆ J of cardinal | σ | ≤ ncε such that(1 − ε ) I n (cid:22) nσ X j ∈ σ ( v j + v ) ⊗ ( v j + v ) (cid:22) (1 + ε ) I n , where v = − | σ | P j ∈ σ v j satisfies k v k ≤ ε √ n .Then, the vector w = v √ nε satisfies k w k ≤ n and therefore by Equation (2),it belongs to conv { v j : j ∈ J } . By Carathodory’s theorem there exist τ ⊆ J with | τ | ≤ n + 1 and ρ i > i ∈ τ such that w = X i ∈ τ ρ i v i , and X i ∈ τ ρ i = 1 . ONTINUOS QUANTITATIVE HELLY-TYPE RESULTS 11
Observe also that − v = | σ | P j ∈ σ v j is in conv { v j : j ∈ σ } . Let T := n | σ | X j ∈ σ v j ⊗ v + n | σ | X j ∈ σ v ⊗ v j + v ⊗ v. As in the proof of Theorem 1.3 it is easy to see that |h T x, x i| ≤ ε for every unitvector x ∈ R n (provided that ε is small enough). Thus(1 − ε ) I n (cid:22) (1 − ε ) I n − T (cid:22) n | σ | X j ∈ σ v j ⊗ v j (cid:22) (1 + ε ) I n − T (cid:22) (1 + 2 ε ) I n . Define X := { v j : j ∈ σ ∪ τ } and E := conv( X ) . Let us show that B n ⊆ cεn / E .Indeed, let x ∈ S n − ; set A := n | σ | P j ∈ σ v j ⊗ v j and ρ := min {h x, v j i : j ∈ σ } .Note that | ρ | ≤ h x, vj i − ρ ≤ j ∈ σ .If ρ < p E ( Ax ) ≤ p E Ax − ρ n | σ | X j ∈ σ v j + p E ρ n | σ | X j ∈ σ v j = p E X j ∈ σ n | σ | ( h x, v j i − ρ ) v j + p E ( nρ ( − v )) ≤ X j ∈ σ n | σ | ( h x, v j i − ρ ) p E ( v j ) − nρp E ( v ) ≤ n (cid:18) √ nε p E ( w ) (cid:19) ≤ c εn / , where we are using that w ∈ K and therefore p E ( w ) ≤ ρ ≥
0, then h x, v j i ≥ j ∈ σ , therefore p E ( Ax ) = p E n | σ | X j ∈ σ h x, v j i v j ≤ n | σ | X j ∈ σ h x, v j i p E ( v j ) ≤ n. This says that p A − ( E ) ( x ) ≤ c εn / for all x ∈ S n − , where c > − ε ) B n ⊆ A ( B n ) ⊆ c (1 + 2 ε ) εn / E. Finally, fix ε := n − δ . Since K is in L¨owner’s position K ⊆ B n ⊆ C ε − ε εn / ⊆ Cn − δ conv( X ) , with | X | = | σ ∪ τ | ≤ cn δ + n + 1 ≤ αn δ . (cid:3) Let us now see the proof of the Theorem 1.4.
Proof.
Consider P := T i ∈ I C i . Without loss of generality we can assume that0 ∈ int ( P ) and that the polar body P ◦ = conv( \ i ∈ I C ◦ i )is in L¨owner’s position. Using Proposition 4.1 for the body K = P ◦ , we know thereexists a set X = { v , · · · , v s } ⊆ P ◦ ∩ S n − such that | X | ≤ αn δ and P ◦ ⊆ Cn − δ conv( X ) , where C > v , · · · , v s are contact points between P ◦ and B n , then we have that v j ∈ T i ∈ I C ◦ i for all j = 1 , · · · , s . This implies thatthere exist s ≤ αn δ and bodies { C i j } , such that v j ∈ C ◦ i j for all j = 1 · · · , s . Thenconv( X ) ⊆ conv( C ◦ i ∪ · · · ∪ C ◦ i s ) and hence P ◦ ⊆ Cn − δ conv( C ◦ i ∪ · · · ∪ C ◦ i s ) . This shows that C i ∩ · · · ∩ C i s ⊆ cn − δ P, and therefore we have the following estimate for the diameterdiam( C i ∩ · · · ∩ C i s ) ≤ cn − δ . This concludes the proof. (cid:3) Final comments: symmetry assumption
It is well-known that if all the bodies are symmetric the bounds for these kind ofresults are better (see, for example, [Bra17a, Theorem 1.2] and [Bra17b, Theorem1.2.]). In that case, for a linear number of convex sets, the bounds are of order n / . One should be tempted to think that relaxing the number of sets in thesestatements provides again stronger estimates but, unfortunately, we cannot havethese type of continuous versions as before. Indeed, the exponent in n in cannotbe improved by allowing more sets: for example we can find w , . . . , w N ∈ S n − (assuming that N is exponential in the dimension n ) such that(4) B n ⊆ N \ j =1 H j ⊆ B n , where H j is defined as the strip H j = { x ∈ R n : |h x, w j i| } . Thus, if s = n δ with δ >
1, for any choice of j , . . . , j s ∈ { , . . . , N } we can use theclassical lower bound for the volume due to Carl-Pajor [CP88] and Gluskin [Glu89],which shows that(5) | H j ∩ · · · ∩ H j s | /n > C p log( n ) . Therefore, if H j ∩ · · · ∩ H j s ⊆ β T Nj =1 H j for some β >
0, by comparing its volumeswe obtain that(6) β > | H j ∩ · · · ∩ H j s | /n | B n | /n > c √ n √ log n , ONTINUOS QUANTITATIVE HELLY-TYPE RESULTS 13 where c >
Acknowledgements
The second author wants to thank A. Giannopoulos who encourage the writingof this manuscript.This research was supported by ANPCyT-PICT-2018-04250 and CONICET-PIP11220130100329CO.
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Departamento de Matem´atica - IMAS-CONICET, Facultad de Cs. Exactas y Natu-rales Pab. I, Universidad de Buenos Aires (1428) Buenos Aires, Argentina
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