A sharp Blaschke Santaló inequality for α-concave functions

Transcription

1 Noname manuscript No will be inserted b the editor) A sharp Blaschke Santaló inequalit for α-concave functions Liran Rotem the date of receipt and acceptance should be inserted later Abstract We define a new transform on α-concave functions which we call the - transform Using this new transform we prove a sharp Blaschke-Santaló inequalit for α-concave functions and characterize the equalit case This extends the known functional Blaschke-Santaló inequalit of Artstein-Avidan Klartag and Milman and strengthens a result of Bobkov Finall we prove that the -transform is a dualit transform when restricted to its image However this transform is neither surjective nor injective on the entire class of α-concave functions Kewords Blaschke-Santaló inequalit convexit α-concavit log-concavit Mathematics Subject Classification 200) 52A40 26B25 Blaschke Santaló tpe inequalities We begin b recalling the classic Blaschke-Santaló inequalit A convex bod in R n is compact convex set with non-empt interior Such a convex bod K is called smmetric if K = K We will denote b K the Lebesgue) volume of K Finall we define the polar bod of K to be K = {x R n : x for all K} where is the standard scalar product on R n Polarit is a basic notion in convex geometr It is eas to see that if K is smmetric convex bod then so is K The map K K satisfies two fundamental properties: It is order reversing: If K K 2 then K K 2 2 It is an involution: For ever smmetric convex bod K we have K = K ) L Rotem School of Mathematical Sciences Tel Aviv Universit Ramat Aviv Tel Aviv Israel liranro@posttauacil

2 2 Liran Rotem These two properties sa that polarit is a dualit transform see eg [2]) The volume of K is related to the volume of K b the Blaschke-Santaló inequalit Theorem Blaschke Santaló) Assume K R n is a smmetric convex bod Then K K D 2 where D R n is the unit Euclidean ball Furthermore we have an equalit if and onl if K is an ellipsoid This theorem was proven b Blaschke in dimensions 2 and 3 and Santaló extended the result to arbitrar dimensions There exists a version of the inequalit for nonsmmetric bodies but for simplicit we will onl deal with the smmetric case The generalized statement proofs and further references can be found eg in [2] In [] Artstein-Avidan Klartag and Milman prove a functional extension of the Blaschke-Santaló inequalit To explain their result and put it in perspective we will need to define α-concave functions: Definition Fix α We sa that a function f : R n [0 ) is α-concave if f is supported on some convex set Ω and for ever x Ω and 0 λ we have f λx + λ)) [λfx) α + λ) f) α ] α We will alwas assume that f is upper semicontinuous and that max fx) = f0) = x Rn this last condition is sometimes known as saing that f is geometric) The class of all such α-concave functions will be denoted b C α R n ) Remember that f : R n [0 ) is called upper semicontinuous if its upper level sets {x R n : fx) t} are closed for all t 0 α-concave functions were first defined b Avriel [5]) and were studied b Borell [8][9]) and b Brascamp and Lieb [0]) In the case α = we understand the definition in the limiting sense as f λx + λ)) max {fx) f)} This just means that f is constant on its support so functions in C R n ) are indicator functions of convex sets which b our assumptions must also be closed and contain the origin If we further assume that f C R n ) satisf 0 < f < then f must be the indicator function of a convex bod and we can identif these functions with the convex bodies themselves If α < α 2 then C α R n ) C α2 R n ) This means that we can think of ever class of functions C α R n ) as extending the class of convex bodies Originall this was done for the class C 0 R n ) of log-concave functions Again we interpret Definition in the limiting sense and sa that a function f : R n [0 ) is log-concave if f λx + λ)) fx) λ f) λ for ever x R n and 0 λ

3 A sharp Blaschke Santaló inequalit for α-concave functions 3 Man definitions and theorems of convex geometr were generalized to the class of log-concave functions Except the usual aspiration for generalit the developed theor helped to prove new deep theorems in convexit and asmptotic geometric analsis For a surve of such results and their importance see [6] One of the first results in this new direction was the functional Santaló theorem In order to state it we will begin with a convenient definition: Definition 2 For ever f C 0 R n ) we define f = exp L log f)) C 0 R n ) where L is the Legendre transform defined b Lϕ) x) = sup R n x ϕ)) This definition just means that if f = e ϕ for a convex function ϕ then f = e Lϕ It turns out that the map f f is a dualit transform on C 0 R n ) and we have the following theorem: Theorem 2 Assume f C 0 R n ) is even and 0 < f < Then f 2 f G) = 2π) n where Gx) = e x 2 2 Equalit occurs if and onl if f = G T for an invertible linear map T : R n R n This inequalit in the even case is originall due to Ball [6]) In [] Artstein- Avidan Klartag and Milman present this result as a Blaschke Santaló tpe inequalit extend the result to the non-even case and characterize the equalit case As a side note let us mention that given our current knowledge the form of Theorem 2 is somewhat surprising Following a series of works b Artstein- Avidan and Milman we now understand that even though the map f f is a dualit transform it is not the correct extension of the classic notion of polarit see [3]) The correct extension is another dualit transform usuall denoted f f which is based on the so-called A-transform Hence we expect the functional Blaschke Santaló inequalit to bound an expression of the form f f ) which is not what we see in Theorem 2 In fact a sharp upper bound on ) is not known even though an asmptotic result was recentl found b Artstein-Avidan and Slomka [4]) The goal of this paper is to discuss Blaschke-Santaló inequalities for α-concave functions for values of α different from 0 The case α > 0 was resolved alread in [] so we will assume from now on that α 0 The following definition appeared in [7]:

4 4 Liran Rotem Definition 3 Assume < α < 0 The convex base of a function f C α R n ) is base α f = f α α Put differentl ϕ = base α f is the unique convex function such that f = + ϕ ) Here and after we use the parameter = α As α 0 and 0 it is often less confusing to use instead of α In the limiting case α = 0 we define base 0 f) = log f Using the notion of a convex base we can extend Definition 2 to the general case: Definition 4 The dual of a function f C α R n ) is the function f C α R n ) defined b relation base α f ) = L base α f)) Note that the operation depends on α Remember that if f C α R n ) then f C α R n ) for all α < α Thinking of f as an element in C α R n ) will ield a different f than thinking about f as function in C α R n ) Therefore strictl speaking we should use a notation like f α However this notation is extremel cumbersome so we will use the simpler notation f and keep in mind the implicit dependence on α We can now state what appears to be the natural extension of Theorem 2 to the α-concave case: Theorem 3 Fix n < α 0 For ever even function f C α R n ) such that 0 < f < we have ) 2 f f H α 2) where H α x) = ) + x 2 2 Cα R n ) This theorem was proven b Bobkov [7]) using a general result b Fradelizi and Meer []) which we will cite in the next section as Theorem 5 The condition α > n is necessar because the function H α is no longer integrable for α n In fact it is not hard to check that H α Hα as α n so it is not possible to find a finite upper bound for f f whenever α n Notice that as α 0 the functions H α converge to the Gaussian G and we obtain Theorem 2 as a special case of Theorem 3 Surprisingl as was alread observed b Bobkov Theorem 3 is not sharp when α < 0 Given the previousl described results it is ver natural to expect an equalit in 2) when f = H α However an explicit et tedious) calculation shows that Hαx) < H α x) for all x 0 so ) 2 H α Hα < H α

5 A sharp Blaschke Santaló inequalit for α-concave functions 5 In fact there exists a unique function G α C α R n ) such that G α = G α and this function is ) G α x) = + x 2 2 Notice that in the limiting case α 0 we have H 0 = G 0 = G but for other values of α these functions are quite different Inspired b Theorems and 2 it ma seem reasonable to conjecture that f ) 2 f G α for all even f C α R n ) Such an inequalit if true will obviousl be sharp Unfortunatel this inequalit is false for α < 0 As one possible counterexample take f = D the indicator function of the ball In this case f = + x ) and a direct computation of the integrals show that this is indeed a counterexample if the dimension n is large enough compared to For concreteness it is enough to take n = for all large enough ) 2 In the next sections we will show the reason inequalit 2) is not sharp is that the transform is not the correct extension of polarit to use in the functional Blaschke-Santaló inequalit In section 2 we will define a new transform on C α R n ) which we call -transform We will then use this -transform to prove a sharp version of Theorem 3 Finall in section 3 we will discuss further properties of which are not directl related to the Blaschke-Santaló inequalit and give a geometric interpretation of this transform 2 A new transform on α-concave functions In [] Artstein-Avidan Klartag and Milman obtain Theorem 2 as the limit of Blaschke-Santaló tpe inequalities for α-concave functions α 0 Let us warn the reader that [] uses a slightl different notation than the one used in this paper: an α-concave function in this paper is the same as a α -concave function in [] and vice versa Inspired b the transforms in [] we define the following: Definition 5 For f C α R n ) we define f x) = inf f) + x ) = sup [f) + x ) ] where the infimum is taken over all points R n such that f) > 0 and x > Like the transform the transform also depends on α so in principle we should write f α Nonetheless we opt for the simpler notation f Let us begin b checking that f C α R n ):

6 6 Liran Rotem Proposition For ever f C α R n ) we have f C α R n ) If f is even so is f Proof For a fixed R n with f) > 0 the function ) f x) = f) + x if x > otherwise is upper semicontinuous and α-concave except the fact it can attain the value + which we usuall exclude from the definition) Now we can write f x) = inf f x) : f)>0 so f is α-concave as the infimum of a famil of α-concave functions Similarl f is upper semicontinuous as the infimum of a famil of upper semicontinuous functions For ever x R n we have Additionall f x) = inf f 0) = inf f) + x f) + 0 ) ) ) = f0) + x0 + = inf f) = sup f) = and we see that f is geometric Hence we have f C α R n ) like we wanted Finall if f is even then f x) = inf = inf so f is even as well f) + x f) + x ) = inf ) = f x) f ) + x Our main goal in this section is to prove the following theorem: Theorem 4 Fix n < α 0 For ever even f C α R n ) such that 0 < f < we have ) 2 f f H α with equalit if and onl if f = H α T for an invertible linear transformation T : R n R n ) Here we have H α x) = + x 2 2 just like in Theorem 3 of Bobkov For α n one cannot hope for a finite upper bound on the product f f This can be seen b choosing f = H α and considering the following proposition: )

7 A sharp Blaschke Santaló inequalit for α-concave functions 7 Proposition 2 H α = H α and H α is the onl function with this propert Proof First we calculate H α explicitl and show that H α = H α B definition H αx) = [ sup H α ) + = sup + 2 ) ] x ) 2 + ) x }{{} ) Notice that if we take a vector and rotate it to have the same direction as x we can onl increase the expression ) Hence H αx) = = sup λ>0 sup λ>0 ) + λx λ x 2 + λ2 x 2 ) 2 2 x λx ) It is now an exercise in calculus to differentiate and check that the supremum is actuall a maximum which is obtained for λ = Hence H αx) = + x 2 + x 2 ) 2 2 = ) + x 2 2 = Hα x) which is what we needed to show Now assume that f C α R n ) is an function such that f = f For ever x R n with fx) > 0 we have fx) = f x) = inf f) + x ) fx) + x 2 so multipling b fx) and taking a square root we get fx) H α x) If fx) = 0 then fx) H α x) holds triviall so the inequalit is true for all x R n It is obvious from the definition that is order reversing so we ma appl it on both sides and obtain so f = H α like we wanted f = f H α = H α Theorem 4 like Theorem 3 will follow from a general result of Fradelizi and Meer []) We state their result here: )

8 8 Liran Rotem Theorem 5 Let f f 2 : R n R + and ρ : R + R + be measurable functions such that f x)f 2 ) ρ 2 x ) for ever x R n such that x > 0 If additionall f is even then f 2 f 2 ρ x )dx) 2 Assume further that ρ is continuous Then equalit will occur if and onl if: ρs)ρt) ρ st) for ever s t 0 2 If n 2 then either ρ0) > 0 or ρ 0 3 There exists a positive definite matrix T and a constant d > 0 such that almost everwhere f x) = d ρ T x 2) f 2 x) = T d ρ x 2) Let us use this result to prove Theorem 4: Proof Proof of Theorem 4) Define a function ρ : R + R + b ρt) = + t ) /2 Fix x R n with x > 0 If fx) = 0 then obviousl fx)f ) ρ 2 x ) If on the other hand fx) > 0 then fx) f ) = inf z = + fx) fz) x ) + z + ) = ρ 2 x ) fx) fx) + x ) From Theorem 5 we conclude that indeed 2 f f ρ x )dx) 2 = H α ) 2 Next we analze the equalit case From Theorem 5 we see that a necessar condition to have equalit is fx) = d ρ T x 2) = d H α T x) for a constant d > 0 and a linear map T : R n R n which we ma take to be positive definite if we want) Since f C α R n ) we know that so we must have f = H α T = f0) = d H α 0) = d = d

9 A sharp Blaschke Santaló inequalit for α-concave functions 9 To see that this condition is also sufficient notice that for ever f C α R n ) and ever invertible linear map T we have f T ) x) = [ sup ft ) + ) ] x = T ) ) x = sup f) + sup f) x T ) + = f T ) x ) Using a simple change of variables and Proposition 2 we get that if f = H α T then f f = H α T ) Hα T ) ) so we are done = dett ) det T ) ) ) 2 H α Hα = H α Remark For simplicit Theorem 4 is onl stated for even functions Fradelizi and Meer also proved in [] a generalization of Theorem 5 for non-even functions which can be used to extend Theorem 4 to the non-even case The proof remains essentiall the same so we leave the details to the interested reader To conclude this section let us compare Theorem 4 with Theorem 3 We have the following proposition: Proposition 3 For ever f C α R n ) we have f f Proof Denote ϕ = base α f) We need to prove that for ever x R n we have f x) f x) which is equivalent to inf + x + ϕ) Choose a sequence { n } such that ) inf + + x n ) f x) + ϕ n) ) x ϕ) We claim it is alwas possible to choose this sequence in such a wa that x n ϕ n ) 0 for ever n Indeed we know that f x) If f x) < we will automaticall have x n > ϕ n ) 0 for large enough n If f x) = just take n = 0 for all n For ever two numbers B A 0 we have + B + A + B A

10 0 Liran Rotem as one easil checks Appling this to B = x n and A = ϕ n) we see that + x n ) + x ) n ϕ n ) f x) + ϕ n) for all n Sending n we see that f x) f x) like we wanted This means that for ever value of α Theorem 3 follows from Theorem 4 When α 0 the transforms and coincide so both theorems reduce to same result - Theorem 2 3 Further properties of -transform In this section we will discuss further properties of the new -transform : C α R n ) C α R n ) for α < 0 We alread used the simple fact that is order reversing: if f g C α R n ) and f g pointwise) then f g Surprisingl however is not a dualit transform as it is not an involution One simple wa of verifing the last assertion is b computing a few examples: Example Let K be a convex bod containing the origin Remember that the gauge function of K is defined b { x } x K = inf r > 0 : r K Let us denote b K the indicator function of K Then a simple calculation gives K = K = + x ) K and [ + x ) ] { } K = min x K In particular we see that K K for ever α < 0 equivalentl for ever < ) In order to prove more delicate properties of the -transform we will need to examine it from a different point view Denote b Cvx 0 R n ) the class of all convex functions ϕ : R n [0 ] such that ϕ is lower semicontinuous and ϕ0) = 0 The map base α : C α R n ) Cvx 0 R n ) from Definition 3 is easil seen to be an order reversing bijection Hence if we wish to understand the -transform it is enough to stud its conjugate T α : Cvx 0 R n ) Cvx 0 R n ) which is defined b T α = base α base α The transform T α can be written down explicitl: Proposition 4 For ever ϕ Cvx 0 R n ) and ever x R n we have x ϕ) x ϕ) T α ϕ) x) = sup = sup 3) R n αϕ) R n + ϕ) )

11 A sharp Blaschke Santaló inequalit for α-concave functions In particular we see that T 0 = L is the Legendre transform This also follows from the fact that on C 0 R n ) the -transform and the -transform coincide Proof Let us use equation 3) as the definition of T α and check that under this definition we reall have ) T α = base α base α This is of course the same as base ) α Tα = base ) α Plugging in all of the definitions we need to prove that for ever ϕ Cvx 0 R n ) and ever x R n + sup ) ) x ϕ) + x = inf + ϕ) ) + ϕ) and checking this equalit involves nothing more than simple algebra Interestingl the transforms T α were introduced and studied b Milman around 970 for ver different applications in functional analsis see [3] [5] for the original papers in Russian and section 33 of the surve [4] for a partial translation to English The remark in the end of section 3 of [] is also relevant but inaccurate) The onl result we will need from these works is the following geometric characterization of T α : Fix a function ϕ Cvx 0 R n ) We will use ϕ to construct a function ρ : R n R [0 ] in the following wa: first we define ρ x ) = + ϕx) Next we extend ρ b requiring it to be -homoegeneous Hence for ever x R n and t 0 we define ) t x ρ x t) = ρ )) = t + ϕ x t = t + t ) x t ϕ t The values of ρ on the hperplane t = 0 are not so important but for concreteness we will define ρx 0) = lim t 0 + ρx t) the limit exists b the convexit of ϕ) The function ρ is -homoegeneous b construction but in general it will not be a norm on R n+ since there is no reason for ρ to be convex Nonetheless we can define the dual norm ρ x + ts x t) = sup s) R ρ s) n+ which is alwas a proper norm on R n+ Now if we restrict ourselves back to the hperplane t = a direct calculation gives ρ x ) = + T αϕ) x) which shows the relation between the transform T α and the classic notion of dualit Using this construction we can prove several properties of the -transform Specificall we have:

12 2 Liran Rotem Theorem 6 Fix < α < 0 and let : C α R n ) C α R n ) be the -transform Then: For ever f C α R n ) we have f f 2 For ever f C α R n ) we have f = f In other words is a dualit transform on its image 3 is neither injective nor surjective Theorem 6 is an immediate corollar of the following proposition establishing the same properties for the transform T α : Proposition 5 Fix < α < 0 and let T α : Cvx 0 R n ) Cvx 0 R n ) be the transform defined above Then: For ever ϕ Cvx 0 R n ) we have T 2 α ϕ ϕ 2 For ever ϕ Cvx 0 R n ) we have T 3 α ϕ = T α ϕ In other words T α is a dualit transform on its image 3 T α is neither injective nor surjective Proof Fix ϕ Cvx 0 R n ) and let ρ : R n+ [0 ] we defined as above It is well known that if ρ is an -homogenous function which is not necessaril convex then ρ ρ In particular + T 2 α ϕ ) x) = ρ x ) ρx ) = + ϕx) which proves ) Since ρ is alread a norm we must have ρ = ρ ) = ρ Restricting again to the hperplane t = we see that T 3 α ϕ = T α ϕ which proves 2) Next we prove 3) and begin b showing that T α is not surjective If ϕ is in the image of T α then the above discussion implies that the corresponding ρ must be a norm on R n+ In particular ρ must be comparable to the Euclidean norm ie there exists a constant C > 0 such that Therefore ρx t) C x t) = C x 2 + t 2 ϕx) = ρ x ) C ) x 2 + C x + and we see that ever function ϕ in the image of T α must grow at most linearl In particular the function ϕx) = x 2 is not in the image of T α so T α is not surjective Finall we will show that T α is also not injective Take an ϕ Cvx 0 R n ) which is not in the image of T α Then T α T 2 α ϕ ) = T 3 α ϕ = T α ϕ even though T 2 α ϕ ϕ This shows that T α is not injective and the proof is complete

13 A sharp Blaschke Santaló inequalit for α-concave functions 3 References Shiri Artstein-Avidan Bo az Klartag and Vitali Milman The Santaló point of a function and a functional form of the Santaló inequalit Mathematika 5-2):33 Februar Shiri Artstein-Avidan and Vitali Milman A characterization of the concept of dualit Electronic Research Announcements in Mathematical Sciences 4: Shiri Artstein-Avidan and Vitali Milman Hidden structures in the class of convex functions and a new dualit transform Journal of the European Mathematical Societ 34): Shiri Artstein-Avidan and Boaz Slomka A note on Santaló inequalit for the polarit transform and its reverse arxiv: Mordecai Avriel r-convex functions Mathematical Programming 2): Februar Keith Ball Isometric problems in l p and sections of convex sets PhD thesis Universit of Cambridge Serge G Bobkov Convex bodies and norms associated to convex measures Probabilit Theor and Related Fields 47-2): March Christer Borell Convex measures on locall convex spaces Arkiv för matematik 2-2): December Christer Borell Convex set functions in d-space Periodica Mathematica Hungarica 62): Herm J Brascamp and Elliott H Lieb On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems including inequalities for log concave functions and with an application to the diffusion equation Journal of Functional Analsis 224): August 976 Matthieu Fradelizi and Mathieu Meer Some functional forms of Blaschke-Santaló inequalit Mathematische Zeitschrift 2562): December Mathieu Meer and Alain Pajor On the Blaschke-Santaló inequalit Archiv der Mathematik 55):82 93 Jul Vitali Milman A certain transformation of convex functions and a dualit of the and δ characteristics of a B-space Doklad Akademii Nauk SSSR 87: Vitali Milman Geometric theor of Banach spaces part II: Geometr of the unit sphere Russian Mathematical Surves 266):79 63 December 97 5 Vitali Milman Dualit of certain geometric characteristics of a Banach space Teorija Funkcii Funkcionalni Analiz i ih Prilozenija 8: Vitali Milman Geometrization of probabilit In Mikhail Kapranov Sergi Kolada Yuri Ivanovich Manin Pieter Moree and Leonid Potagailo editors Geometr and Dnamics of Groups and Spaces volume 265 of Progress in Mathematics pages Birkhäuser Basel Liran Rotem Support functions and mean width for α-concave functions Advances in Mathematics 243:68 86 August 203