-concave functions and a functional extension of mixed volumes
In this paper we define an addition operation on the class of quasi-concave functions. While the new operation is similar to the well-known sup-convolution, it has the property that it polarizes the Lebesgue integral. This allows us to define mixed integrals, which are the functional analogs of the classic mixed volumes.
We extend various classic inequalities, such as the Brunn-Minkowski and the Alexandrov-Fenchel inequality, to the functional setting. For general quasi-concave functions, this is done by restating those results in the language of rearrangement inequalities. Restricting ourselves to log-concave functions, we prove generalizations of the Alexandrov inequalities in a more familiar form.
Mixed volumes, which are the polarization of volume with respect to the Minkowski addition, are fundamental objects in convexity. In this note we announce the construction of mixed integrals, which are functional analogs of mixed volumes. We build a natural addition operation on the class of quasi-concave functions, such that every class of -concave functions is closed under . We then define the mixed integrals, which are the polarization of the integral with respect to .
We proceed to discuss the extension of various classic inequalities to the functional setting. For general quasi-concave functions, this is done by restating those results in the language of rearrangement inequalities. Restricting ourselves to -concave functions, we state a generalization of the Alexandrov inequalities in their more familiar form.
Key words and phrases:mixed integrals, -concavity, quasi-concavity, mixed volumes, log-concavity, Brunn-Minkowski.
2000 Mathematics Subject Classification:52A39, 26B25
1. -concave functions
Let us begin by introducing our main objects of study:
Fix . We say that a function is -concave if is supported on some convex set , and for every and we have
For simplicity, we will always assume that is upper semicontinuous, and as . The class of all such -concave functions will be denoted by .
In the above definition, we follow the convention set by Brascamp and Lieb (), but the notion of -concavity may be traced back to Avriel () and Borell (, ). Discussions of -concave functions from a geometric point of view may be found, e.g., in  and .
In the cases we understand Definition 1 in the limit sense. For example, if is supported on some convex set , and
for all and . Of course, this just means that is constant on . In other words, we have a natural correspondence between to the class of compact, convex sets in : every function is of the form
for some .
Notice that if then (see ) . Therefore, we can view the class for as an extension of the class of convex sets. Our main goal in this note is to extend the geometric notion of mixed volumes from to the different classes of -concave functions.
For convenience, we will restrict ourselves to the case . For , it is easy to see that is -concave if and only if is a convex function on . The cases are important, and deserve a special name:
A -concave function is called log-concave. These are the functions such that
for all and . We will usually write instead of .
A -concave function is called quasi-concave. These are the functions such that
for all and . We will usually write instead of .
We will now see that if , there is a natural correspondence between and convex functions on . Since we only care about negative values of , it will sometimes be convenient to use the parameter . The following definition appeared in :
The convex base of a function is
Put differently, is the unique convex function such that
In the limiting case we define .
By our assumptions on , the function is convex, lower semicontinuous, with and as . We will denote this class of convex functions by (this is not an entirely standard notation), and notice that the map is a bijection between and . It follows immediately, for example, that every function is continuous on its support, because the same is true for convex functions.
In the case , we have no such correspondence. It is therefore not surprising that it possible to construct quasi-concave functions which are not continuous on their support. Indeed, fix convex sets and define .
Remember that if , then for every . However, in general we have , so the base depends on the class we choose to work in, and not only on our function . However, in the specific case for some convex set , we have
for every value of .
On there is a natural addition operation, known as inf-convolution:
For we define their inf-convolution to be
Similarly, if and we will define
The definition of was chosen to have , as one easily verifies. It is also easy to see that we have commutativity, associativity and distributivity.
We will not explain the exact sense in which these operations are natural, and instead refer the reader to the first section of . We will note, however, that Definition 4 extends the classical operations on convex bodies: If and then
Here is the Minkowski sum of convex bodies, defined by
and is defined by
We will now define addition on , using the established correspondence between and :
Fix . Then:
For we define their -sum by the relation
For and we define via the relation
Again, the definition of -sum depends on , and not only on and : If and , then in general we have . However, for indicators of convex sets we have
for all .
The definition of -sum may be written down explicitly, without referring to the convex bases. For we have
and for we get the limiting case
The operation on is known as Asplund-sum, or sup-convolution (see, e.g., ).
For we cannot define using the same approach as Definition 5, because we do not have the notion of a base for quasi-concave functions. However, we may use equation 1.1, and the fact that for every we have
This, and a similar consideration for , leads us to define:
For we define their quasi-sum by
For and we define by
For , we explicitly define
This definition ensures that for every . We use the notations and instead of and because these operations will play a fundamental role in the rest of this paper.
So far we discussed properties of -concave functions which made sense for every value of . We now want to state a few results that are only true for quasi-concave functions and quasi-sums. We will need the following definition:
For a function and we define
to be the upper level sets of .
We now have the following result, which will play an important role in this note:
Fix . Then if and only if are compact, convex sets for all .
For every , and we have
The sum has another important property, we would now like to discuss. Remember that if , then for all , so we may look at the function . Generally, the function does not have to be in , even though and are. Let us consider an example: choose , and choose . In this case we have
and since we have
and it is easy to check that . Of course, we must have .
It turns out that such a situation cannot happen when :
If for some , so does .
The proofs of the last two results will appear in . Notice that by this theorem we have two different addition operations on . One is , and the second is the “universal” .
2. Mixed integrals
Recall the following theorem by Minkowski (see, e.g.  for a proof):
Fix . Then the function , defined by
is a homogenous polynomial of degree , with non-negative coefficients.
The coefficients of this polynomial are called mixed volumes. To be more exact, we have a function
which is multilinear (with respect to the Minkowski sum), symmetric (i.e. invariant to a permutation of its arguments), and which satisfies . From these properties it is easy to deduce that
The number is called the mixed volume of the .
As we stated before, our goal is to prove a functional extension of Minkowski’s theorem, and to define a functional extension of mixed volumes. We will state our results on , since this is the largest class of functions we consider, so all the results will be true for every class (and, in particular, on ). Of course, in order to formulate and prove such a theorem, we need to decide what are the functional analogs of volume, and of Minkowski sum.
For volume, if we want our theorem to be a true extension of Minkowski’s, we need a functional on such that . A natural candidate is the Lebesgue integral,
For the extension of addition, it turns out that the best possibility is the quasi-sum . In fact, we have the following theorem:
Fix . Then the function , defined by
is a homogenous polynomial of degree , with non-negative coefficients.
The proof will appear in . In complete analogy with the case of convex bodies, this theorem is equivalent to the existence of a function
which is symmetric, multilinear (with respect to , of course) and satisfies . We will call the number the mixed integral of . In other words, the mixed integral is the polarization of the integral . We have the following representation formula for the mixed integrals:
Fix . Then
Mixed integrals share many important properties with the classical mixed volumes. We will mention a few in the following theorem:
For we have
For every we have . More generally, if we also have such that for all , then
is rotation and translation invariant. Also, if we define
for and , then
Let and denote by the support of . Then if and only if .
Fix an integer and functions . Then the functional
satisfies a valuation type property: if and as well, then
Here is an alternative notation for .
Denote , and define by
notice that is non-negative and non-increasing, by monotonicity of mixed volumes. Since
(the bar denotes the topological closure), and by continuity of mixed volumes, we have .
Therefore, if then , so . If, on the other hand, , then for all smaller than some , so
A particularly interesting example of mixed volumes is quermassintegrals. For , we define the -th quermassintegral to be
where is the Euclidean unit ball. Similarly, for we will define
By checking the definitions, we see that if and is a convex set, then
for every . In particular, the left hand side is independent of the exact value . Since for we obtain a polynomial in , we must obtain the same polynomial for every value of . Therefore, as a direct corollary of Theorem 10 we obtain the following statement, which was independently obtained by Bobkov, Colesanti and Fragalà (see ):
Fix . For and , define
Then we have
As stated, this result was also discovered by Bobkov, Colesanti and Fragalà. Their paper continues to prove several properties of the quermassintegrals, such as Prékopa-Leindler inequalities and a Cauchy-Kubota integral formula. We will not pursue these points in this note. Let us stress that Proposition 13 only works for quermassintegrals, where the different notions of sum happen to coincide. For general mixed integrals, it is impossible to get polynomiality for the operation unless .
Now that we have a functional version of Minkowski’s theorem, we would like to prove inequalities between different mixed integrals. Let us use the isoperimetric inequality as a test case. The classical isoperimetric inequality, arguably the most famous inequality in geometry, claims that for every (say convex) body we have
Here is the surface area of , defined by
We would like to generalize this result to the functional setting. The naive approach would be the try and bound from below using . Unfortunately, this is impossible to do for general quasi-concave functions. In fact, it is possible to construct a sequence of functions such that but as . We will present a concrete example in .
Thus we will use a different approach, and prove an extension of the isoperimetric inequality by recasting it as a rearrangement inequality. In order to explain this idea, consider the following definition
For a compact , define
In other words, is the Euclidean ball with the same volume as .
For , define its symmetric decreasing rearrangement using the relation
It is easy to see that this definition really defines a unique function , which is rotation invariant.
Now, the isoperimetric inequality may be restated as for . In this formulation, the functional extension turns out to be true:
If , then , with equality if and only if is rotation invariant.
This inequality is indeed an extension of the isoperimetric inequality, as can be seen by choosing . It can also be useful for general quasi-concave functions, because it reduces an -dimensional problem to a 1-dimensional one – the function is rotation invariant, and hence essentially “one dimensional”. However, we stress again that in general, this inequality does not yield a lower bound for in terms of , as such a bound is impossible.
Many other inequalities can be extended using similar formulations. For example, the Brunn-Minkowski inequality states that for every (say convex) sets we have
Again, in general, it is impossible to bound from below using and . However, the Brunn-Minkowski inequality may be written as , and in this representation it generalizes well:
For every we have
In  we will prove the above two results, as well as extensions of the generalized Brunn-Minkowski inequalities for mixed volumes and the Alexandrov-Fenchel inequality. We will not describe these results here, since they require the notion of a “generalized rearrangement”. Instead, let us mention one elegant corollary of the Alexandrov-Fenchel inequality. Remember that for convex bodies we have the inequality
The functional analog of this result is the following inequality:
For all functions we have
If one is willing to restrict oneself to some class of -concave functions, then it is suddenly possible to prove inequalities between mixed integrals in a more familiar form. In order to state the result, let us define for every a function by
where, as usual . Put differently, we choose to satisfy (for , we obtain ) . By abuse of notation, we will also think of as the function from to defined by
so . We are now ready to state:
Fix a function and integers and such that . Then we have
assuming . Equality occurs if and only if for some .
Of course, if the theorem is either trivial (if ) or meaningless (if ). The condition is equivalent to , and implies that all the other quantities in the theorem are finite as well. Notice that since are all integers, we have . Hence we need to choose for the theorem to have any content.
In , a proof will be given for the case , where the condition is true for all . A key ingredient in the proof is a bound on the growth of moments of log-concave functions. The general proof is similar, and depends on the following lemma:
Let be an -concave function such that . Then for every we have
with equality if and only if
for some .
Again, the condition simply ensures that all of the integrals are finite. Under this condition the lemma follows from Lemma 4.2 of , by taking (In  the equality condition is not explicitly stated, but it can be deduced by carefully inspecting the proof).
First, we reduce the general case to the rotation invariant case, by replacing with some generalized rearrangement . The definition of and its necessary properties will appear in .
So, assume without loss of generality that is rotation invariant. By abuse of notation we will write . A direct computation (to also appear in ) gives
and similarly for . Remember that we assumed , which is the same as
This implies that , or , like we claimed. In particular we have
so we can use Lemma 19 and conclude that
This is the same as
which is what we wanted.
The equality case will follow from the equality case of Lemma 19, but we will not give the details here. ∎
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