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 69978 Tel Aviv Israel E-mail: liranro@posttauacil
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
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 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
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 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: )
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 2 2 + + 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 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
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
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) )
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:
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
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