Support functions and mean width for \alpha-concave functions

Support functions and mean width for -concave functions

Liran Rotem School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, 69978, Tel Aviv, Israel liranro1@post.tau.ac.il
Abstract.

In this paper we extend some notions, previously defined for log-concave functions, to the larger domain of so-called -concave functions. We begin with a detailed discussion of support functions – first for log-concave functions, and then for general -concave functions. We continue by defining mean width, and proving some basic results such as an Urysohn type inequality. Finally, we demonstrate how such geometric results can imply Poincaré type inequalities.

Key words and phrases:
-concavity, mean width, support function, Urysohn

1. Support functions and -concave functions

A well known construction in classic convexity is the support function of a convex body. Let be a closed, convex set. Then its support function is a function , defined by

where denotes the standard Euclidean structure on .

To begin our discussion, we will briefly state a few basic properties of support functions. A more detailed account, together with all of the relevant proofs, can be found for example in section 1.7 of [12]. Define

For every its support function belongs to

Furthermore, the map sending to has the following properties:

  1. is bijective.

  2. is order preserving: if and only if (here and after, means for all )

  3. is additive: For every we have , where the addition on the left hand side is Minkowski addition:

It turns out that these properties suffice to characterize uniquely, up to a linear change of variables. In fact, one can do even better: From the work of Gruber in [9] it is easy to deduce the following:

Theorem.

Assume satisfies (P1) and (P2) for . Then there exists an invertible affine map such that for all .

Similarly, In [2] Artstein-Avidan and Milman prove:

Theorem.

Assume satisfies (P1) and (P3) for . Then there exists an invertible linear map such that for all .

We will now shift our attention from bodies to functions. In recent years, many notions and results from convexity were generalized from the class of convex bodies to larger domains. One usual choice for such a domain is the class of log-concave functions. To give an exact definition, we can define

and then the class of log-concave functions is simply

Put differently, a function is log-concave if is a convex function. We always assume our log-concave functions are upper semicontinuous, and explicitly exclude the constant function . There is a natural embedding of into , which maps every convex body to its characteristic function

Some notions from convexity extend easily to the class of log-concave functions. For example, since

one can say that the integral of a function extends the notion of the volume of a convex body . Other extensions might not be as obvious. It is by now standard to extend the notion of Minkowski addition by the operation known as sup-convolution, or Asplund sum: For we define their sum by

Notice that this is indeed a generalization, in the sense that .

Similarly, it is standard to extend the notion of support function by defining

for The here denotes the classic Legendre transform, defined by

Again, this is a proper generalization, in the sense that , as one easily checks.

It turns out that support functions of log-concave functions share most of the important properties of support functions of convex bodies. More specifically, the function mapping to has the following properties:

  1. is bijective.

  2. is order preserving: if and only if .

  3. is additive: For every we have .

Again, one can use properties (Q1)-(Q3) to uniquely characterize the support function up to a linear change of variables. In [1] Artstein-Avidan and Milman prove

Theorem.

Assume satisfies (Q1) and (Q2). Then there exists an invertible affine map , constants and a vector such that

for all .

Additionally, in [2] the same authors prove the following:

Theorem.

Assume satisfies (Q1) and (Q3). Then there exists an invertible affine map and a constant such that

for all .

The last two theorems actually serve an important purpose. The definition of the support function of a convex body exists for a long time, and has proven itself to be extremely useful. The definition of the support function of a log-concave function, however, is much newer, and it is reasonable to debate the question of whether this definition is the “right” one. These theorems give us a way to justify our definitions: If, for example, one believes that (Q1) and (Q2) are reasonable properties that any definition will have to satisfy, then the exact definition follows immediately.

However, as was pointed out by Vitali Milman, the assumption (Q1) may not be as innocent as it first appears. Injectivity of is fairly natural, and follows easily from property (Q2) as well. Surjectivity, on the other hand, is a more delicate matter. After all, the support function of a convex body is not an arbitrary convex function, but always a positively homogenous one. It is possible that should also be a proper subset of , and that in order to get every function as a support function one should increase the domain even further.

As it turns out, there is a well known way to extend to a larger class of functions. Consider the following definition:

Definition 1.

Fix . We say that a function is -concave if is supported on some convex set , and for every and we have

For we understand this inequality in the limit sense. This means that is -concave if

is -concave if it is log-concave:

and -concave if

which implies that is constant on .

In this definition we follow the conventions of Brascamp and Lieb in [6], but the notion can be traced to the works of Avriel [3] and Borell [5]. The interested reader may also consult [4] for applications more similar in spirit to this paper.

One of the goals of this paper is to demonstrate how some constructions, usually carried out for log-concave functions, can also be carried out for general -concave functions. These constructions will include the support function, and the mean width. This discussion, together with a more systematic treatment of -concave functions, will appear in section 3. For now, let us just mention the fact that if , then every -concave function, is also -concave. We can use this fact to generate the following example:

Example 2.

Fix . The claim that is -concave is equivalent to saying that is convex. This means we can write every -concave function, and hence every log-concave function, as

for some convex function (notice that is positive). Define by

It is easy to verify that extends the classic support function, in the sense that if then . It is equally easy to check that this satisfies property (Q2), but not property (Q1). Since is in general very different from , we see that the Artstein-Milman characterization theorem fails completely without the surjectivity assumption.

To gain an insight into the origins of this example, notice that as we have , so , at least on a heuristic level. Intuitively, one may think of as the “right” definition of the support function of an -concave function, and the standard definition is just the special case . We will revisit this point of view in section 3.

It is interesting to note that the above example does not satisfy property (Q3), so at least in this sense it is less natural then the standard construction. Vitali Milman asked whether it possible to assume both (Q2) and (Q3), and prove a characterization theorem which does not require surjectivity. Indeed, this is the case:

Theorem 3.

Assume we are given a function and an operation with the following properties:

  1. is order preserving: if and only if .

  2. extends the usual support functional: If , then .

  3. .

Then we must have

for some , and .

Note that even though we do not assume surjectivity a priori, it follows a posteriori that must be onto . Another interesting feature of the theorem is that our third assumption is somewhat weaker than (Q3), as we only need to assume that is additive with respect to some addition operation . Therefore this theorem characterizes not only the support function, but the sup-convolution operation as well.

Theorem 3 will follow easily from the following theorem:

Theorem 4.

Assume a function satisfies the following:

  1. is order preserving: if and only if .

  2. If is a positively homogenous function then

  3. The set is closed under pointwise addition.

Then for some .

Proof of the reduction.

Assume theorem 4 holds. In order to prove theorem 3, choose satisfying all of the assumptions. Define by

It it easy to see that satisfies all of the assumptions of theorem 4, so

for some . For every define , and notice that . Hence we get

like we wanted. In particular, for every we will get

and since is injective . This completes the proof. ∎

The rest of this paper is organized as follows: Section 2 will include the proof of theorem 4. The proof is rather long, and is composed of several independent ingredients. Some of the ingredients are fairly standard by now, but some are probably new. Let me thank Alexander Segal and Boaz Slomka for providing some of these arguments. Section 3 will be devoted to -concave functions. We will extend the notions discussed in this section, specifically addition and support functions, to the realm of -concave functions. Finally, we will define the mean width of an -concave function, and generalize some known results to the new setting.

2. Proof of theorem 4

We will now prove Theorem 4. As the proof is quite long, we will divide it into several parts:

1. “Negatively affine” functions

Fix a vector . The linear function is 1-homogenous, so . We will now deal with affine functions of the form

for some (we call such functions “negatively affine functions”). Since and is order preserving we must have , and it follows immediately that

for some constant .

Later we will show that contains all affine functions. For now, just notice that if then we can write

and since is closed under addition we get that . In particular, there exists a sequence such that .

Finally, notice that if a negatively affine function

is in , then we can apply the same reasoning as above for and conclude that for some .

2. “Positively affine” functions

Assume now that we are given a function with and . Define , and let

be any tangent to . As we saw before, we can choose a constant such that

Since , it follows that , where

and then we must have . In other words, every tangent to is of the form for some , and this can only happen if

for some . Since we of course have .

Again, if for happens to be in , we can repeat the argument in reverse and conclude that

for some .

3. Surjectivity on affine functions

So far we have seen that that image of any affine function is affine. We will now show that all affine functions are in . Fix , and define according to the formula

Also define

Notice that is closed under addition: If then

so as well. We will now apply the following Lemma:

Lemma 5.

Assume is a subset with the following properties:

  1. If then .

  2. There exists such that .

  3. There exists such that .

Then is either a cyclic subgroup of or dense in .

This result, or slight variations thereof, is well known in some fields. For the sake of completeness, we will prove Lemma 5 after we finish proving Theorem 4.

By the above discussion we see that satisfies the hypotheses of Lemma 5, so it is either cyclic or dense. Since is injective, must is uncountable, and since cyclic groups are all countable, must be dense. Now is a monotone function with dense image, and it is an easy exercise that all such functions are onto. Therefore and we proved the desired result.

4. Delta functions

We will return to affine functions shortly, but before we do we need to discuss delta functions. For and define

Our goal is to show that delta functions are mapped to delta functions. Assume by contradiction that is not a delta function, so there exists such that . We will divide the proof into two cases:

  • There exists a constant such that . Without loss of generality we can assume (otherwise, swap and in the following proof). Define two functions

    (where denotes the euclidean norm). as a positively homogenous function, and as a constant, hence affine, function. Therefore if we define for then as well. Define . Since , we have , so for some . In particular, we must have or vice versa. But this implies that are comparable as well, which is a contradiction:

  • Now assume and are not on the same ray. In this case define

    and the rest of the proof is exactly the same as the previous case.

In both cases we arrived at a contradiction, so we proved that indeed is a delta function like we wanted.

5. Tangents

Let be an affine function and be arbitrary. We say that is tangent to if , but for every . It is well known that

Our simple claim is that is tangent to if and only if is tangent to . Indeed, if is tangent to , then we immediately get . If for some then is an affine function such that

which is impossible. The other direction is proven in exactly the same way.

In particular, if , then is tangent to if and only if . Therefore if is an affine function and , we can always find such that

6. Collinearity

We will identify every affine map with the point . Under this identification our map induces a bijection defined by

Our current goal is to show that maps collinear points into collinear points. The following lemma will prove itself useful:

Lemma 6.

Assume are 3 affine functions such that and are not parallel. Then are collinear if and only if whenever we also have .

Again, we will postpone the proof of Lemma 6 until the end of the proof of Theorem 4. For now we will use this lemma to prove the result: Assume are collinear affine functions, and define . If and are parallel then all six functions are parallel to each other and there is nothing to prove.

In the general case, assume is any point such that

and define . We’ve already seen that in this case we must have

where . Since are collinear we get from Lemma 6 that as well. This implies that is tangent to , so is tangent to and . Again by Lemma 6 we get that are collinear like we wanted.

By the fundamental theorem of affine geometry, it now follows that is affine, so is affine as well (for an exact formulation of the fundamental theorem and a sketch of the proof, the reader may consult [1]). This means that we can write

for some constants and (which are of course independent of and ).

We know that , so for all , which implies . Also, for to be order preserving, we must have .

7. Finishing the proof

We now know that

for some . Remember that in the statement of Theorem 4 we had one degree of freedom - we don’t want to prove that , but that for some . We will now use this degree of freedom and assume that (formally, this means we replace with defined by . We will keep using the notation for the new function).

For every function and any affine , we know that is tangent to if and only if is tangent to . In other words, and have exactly the same tangents, so and our proof is finally complete.

8. Proofs of the lemmas

Proof of Lemma 5..

First we note that it is enough to prove that is either cyclic or dense in . Indeed, assume that is dense in . We know that there exists an element in . If is any number, we can choose so large that . Now we can find a sequence such that , and then , so is dense in . Of course, by a symmetric argument, it is also enough to prove that is dense in .

Now we define

If then there exists a sequence such that for all and . But then the set is dense in , so we are done. Similarly, if then is dense in and we are done as well. Hence we will assume that , and prove that is cyclic.

Our next goal is to prove that . If not, we may assume without loss of generality that , or, put differently, . Choose sequences such that and . Then . Since , for large enough we have , which is a contradiction to the definition of . Therefore like we wanted.

Now we prove that . If , then for every one can find an element such that . In particular, we can choose such that . Like before, choose a sequence such that . Then , and in particular for large enough we have . Again, this is a contradiction to the definition of , so . An identical argument shows that as well.

To conclude the proof we finally show that . The fact that is now obvious. For the other direction, every element can be written as for and . If then

as well, so by the minimality of we must have . Therefore , so . A similar argument works in the case ,and the proof is complete. ∎

Proof of Lemma 6..

One of the implications is simple: If are collinear then we can write

for some . This implies that

and then implies as well.

The other implication is almost as easy. Write

for . Our assumption can be reformulated as saying that implies By standard linear algebra, this can only happen if there exists a such that

This is equivalent to

which implies collinearity of like we wanted. ∎

3. Mean width for -concave functions

We will now begin our discussion of -concave functions (see Definition 1). For simplicity, we will restrict ourselves to case . For any such , define

For example, we have . As stated in section 1, we have whenever .

Remark 7.

In [5], Borell defines not only -concave functions, but also the notion of a -concave measure. A Radon measure on is -concave if for any non empty Borel sets and any we have

Borell then proves that -concave functions and -concave measures are closely related: Assume is not support on any hyperplane. Then is -concave if and only if , is absolutely continuous with respect to the Lebesgue measure, and the density is -concave, for .

Notice that for such a density , we must have , so some authors only discuss -concave functions for such values of . We will need the assumption for some theorems, but other results will hold in full generality.

Since we only care about negative values of , it will often be more convenient, and less confusing, to use the parameter . For example, we will use the new notation in the following definition:

Definition 8.

The convex base of the a function is

Put differently, is the unique convex function such that

The above definition is inspired by the work of Bobkov in [4]. While the definition might seem unintuitive at first, it has a couple of appealing features :

  • In the limiting case (), we get the relation . This is the standard and often used bijective, order reversing map between and .

  • If is an indicator function, then

    is the well known “convex indicator function” of . In particular, is independent of in that case.

Notice, however, that unlike the log-concave case, the map is not surjective, as we always have .

If we are willing to treat as the proper generalization of to the -concave case, a few important definitions emerge immediately:

Definition 9.
  1. The support function of a function is

  2. The sum of two functions is defined by

    assuming the right hand side is a indeed a convex base for an -concave function (see the discussion above Proposition 10). Here is the standard inf-convolution, defined by

  3. If and , the -homothety of defined by

When there is no cause for confusion, we will omit the script and write , , and .

The above definitions were constructed to interact well with one another. We have for example

as well as and other similar equalities. However, one should be aware of two important caveats.

The first thing to observe is that the above definitions really depend on . We know that if , then for every . Nonetheless, we usually have , and similarly for additions and homotheties. An important exception to this rule is the case of indicator functions. If and , then , and , for all values of .

The second, technical, caveat is that additions and homotheties are not always defined. If, for example, , then

But this is impossible, since for every we have . Addition is defined, however, under some mild conditions on and (for example it is enough to assume ). A particularly nice case is the case of convex combinations, where we have the following simple formula:

Proposition 10.

Fix and choose . Define

Then

Proof.

This is nothing more than an explicit calculation. Denote and . Then:

and raising both sides to power we get the result. ∎

Proposition 10 is especially useful when combined with a known inequality, discovered independently by Borell, ([5]) and Brascamp and Lieb ([6]):

Theorem (Borell-Brascamp-Lieb).

Assume we are given measurable functions and numbers , such that

whenever . Then

where .

The important of the parameter was explained in Remark 7. Notice that when we get and the theorem reduces to the Brunn-Minkowski theorem. When we get that as well and the theorem reduces to the a special case known as the Prékopa–Leindler inequality.

From Proposition 10 and Theorem 3 we immediately get:

Corollary 11.

If and , then

Our next goal is to define the mean width of an -concave function. For log-concave functions, the concept of mean width was originally defined by Klartag and Milman in [10]. If , the Klartag-Milman definition for the mean width of is, up to some universal constant,

where is the (unnormalized) Gaussian. Since we are dealing with log-concave functions, means in our notation. In [11], the author presented an equivalent definition, as the average of the support function with respect to the Gaussian measure:

Both definitions can be extended, mutatis mutandis, to general -concave functions.

Definition 12.
  1. For define a function by

    where, as usual . In other words, we choose to satisfy .

  2. For we define its - mean width as