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In one of the earliest posts on this blog, I talked about the ability to “arbitrage” a disparity of symmetry in an inequality, and in particular to “amplify” such an inequality into a stronger one. (The principle can apply to other mathematical statements than inequalities, with the “hypothesis” and “conclusion” of that statement generally playing the role of the “right-hand side” and “left-hand side” of an inequality, but for sake of discussion I will restrict attention here to inequalities.) One can formalise this principle as follows. Many inequalities in analysis can be expressed in the form
for all in some space
(in many cases
will be a function space, and
a function in that space), where
and
are some functionals of
(that is to say, real-valued functions of
). For instance,
might be some function space norm of
(e.g. an
norm), and
might be some function space norm of some transform of
. In addition, we assume we have some group
of symmetries
acting on the underlying space. For instance, if
is a space of functions on some spatial domain, the group might consist of translations (e.g.
for some shift
), or perhaps dilations with some normalisation (e.g.
for some dilation factor
and some normalisation exponent
, which can be thought of as the dimensionality of length one is assigning to
). If we have
for all symmetries and all
, we say that
is invariant with respect to the symmetries in
; otherwise, it is not.
Suppose we know that the inequality (1) holds for all , but that there is an imbalance of symmetry: either
is
-invariant and
is not, or vice versa. Suppose first that
is
-invariant and
is not. Substituting
by
in (1) and taking infima, we can then amplify (1) to the stronger inequality
In particular, it is often the case that there is a way to send off to infinity in such a way that the functional
has a limit
, in which case we obtain the amplification
of (1). Note that these amplified inequalities will now be -invariant on both sides (assuming that the way in which we take limits as
is itself
-invariant, which it often is in practice). Similarly, if
is
-invariant but
is not, we may instead amplify (1) to
and in particular (if has a limit
as
)
If neither nor
has a
-symmetry, one can still use the
-symmetry by replacing
by
and taking a limit to conclude that
though now this inequality is not obviously stronger than the original inequality (1) (for instance it could well be trivial). In some cases one can also average over instead of taking a limit as
, thus averaging a non-invariant inequality into an invariant one.
As discussed in the previous post, this use of amplification gives rise to a general principle about inequalities: the most efficient inequalities are those in which the left-hand side and right-hand side enjoy the same symmetries. It is certainly possible to have true inequalities that have an imbalance of symmetry, but as shown above, such inequalities can always be amplified to more efficient and more symmetric inequalities. In the case when limits such as and
exist, the limiting functionals
and
are often simpler in form, or more tractable analytically, than their non-limiting counterparts
and
(this is one of the main reasons why we take limits at infinity in the first place!), and so in many applications there is really no reason to use the weaker and more complicated inequality (1), when stronger, simpler, and more symmetric inequalities such as (2), (3) are available. Among other things, this explains why many of the most useful and natural inequalities one sees in analysis are dimensionally consistent.
One often tries to prove inequalities (1) by directly chaining together simpler inequalities. For instance, one might attempt to prove (1) by by first bounding by some auxiliary quantity
, and then bounding
by
, thus obtaining (1) by chaining together two inequalities
A variant of the above principle then asserts that when proving inequalities by such direct methods, one should, whenever possible, try to maintain the symmetries that are present in both sides of the inequality. Why? Well, suppose that we ignored this principle and tried to prove (1) by establishing (4) for some that is not
-invariant. Assuming for sake of argument that (4) were actually true, we could amplify the first half
of this inequality to conclude that
and also amplify the second half of the inequality to conclude that
and hence (4) amplifies to
Let’s say for sake of argument that all the quantities involved here are positive numbers (which is often the case in analysis). Then we see in particular that
Informally, (6) asserts that in order for the strategy (4) used to prove (1) to work, the extent to which fails to be
-invariant cannot exceed the amount of “room” present in (1). In particular, when dealing with those “extremal”
for which the left and right-hand sides of (1) are comparable to each other, one can only have a bounded amount of non-
-invariance in the functional
. If
fails so badly to be
-invariant that one does not expect the left-hand side of (6) to be at all bounded in such extremal situations, then the strategy of proving (1) using the intermediate quantity
is doomed to failure – even if one has already produced some clever proof of one of the two inequalities
or
needed to make this strategy work. And even if it did work, one could amplify (4) to a simpler inequality
(assuming that the appropriate limit existed) which would likely also be easier to prove (one can take whatever proofs one had in mind of the inequalities in (4), conjugate them by
, and take a limit as
to extract a proof of (7)).
Here are some simple (but somewhat contrived) examples to illustrate these points. Suppose one wishes to prove the inequality
for all . Both sides of this inequality are invariant with respect to interchanging
with
, so the principle suggests that when proving this inequality directly, one should only use sub-inequalities that are also invariant with respect to this interchange. However, in this particular case there is enough “room” in the inequality that it is possible (though somewhat unnatural) to violate this principle. For instance, one could decide (for whatever reason) to start with the inequality
to conclude that
and then use the obvious inequality to conclude the proof. Here, the intermediate quantity
is not invariant with respect to interchange of
and
, but the failure is fairly mild (changing
and
only modifies the quantity
by a multiplicative factor of
at most), and disappears completely in the most extremal case
, which helps explain why one could get away with using this quantity in the proof here. But it would be significantly harder (though still not impossible) to use non-symmetric intermediaries to prove the sharp version
of (8) (that is to say, the arithmetic mean-geometric mean inequality). Try it!
Similarly, consider the task of proving the triangle inequality
for complex numbers . One could try to leverage the triangle inequality
for real numbers by using the crude estimate
and then use the real triangle inequality to obtain
and
and then finally use the inequalities
but when one puts this all together at the end of the day, one loses a factor of two:
One can “blame” this loss on the fact that while the original inequality (9) was invariant with respect to phase rotation , the intermediate expressions we tried to use when proving it were not, leading to inefficient estimates. One can try to be smarter than this by using Pythagoras’ theorem
; this reduces the loss from
to
but does not eliminate it completely, which is to be expected as one is still using non-invariant estimates in the proof. But one can remove the loss completely by using amplification; see the previous blog post for details (we also give a reformulation of this amplification below).
Here is a slight variant of the above example. Suppose that you had just learned in class to prove the triangle inequality
for (say) real square-summable sequences ,
, and was tasked to conclude the corresponding inequality
for doubly infinite square-summable sequences . The quickest way to do this is of course to exploit a bijection between the natural numbers
and the integers, but let us say for sake of argument that one was unaware of such a bijection. One could then proceed instead by splitting the integers into the positive integers and the non-positive integers, and use (12) on each component separately; this is very similar to the strategy of proving (9) by splitting a complex number into real and imaginary parts, and will similarly lose a factor of
or
. In this case, one can “blame” this loss on the abandonment of translation invariance: both sides of the inequality (13) are invariant with respect to shifting the sequences
,
by some shift
to arrive at
, but the intermediate quantities caused by splitting the integers into two subsets are not invariant. Another way of thinking about this is that the splitting of the integers gives a privileged role to the origin
, whereas the inequality (13) treats all values of
equally thanks to the translation invariance, and so using such a splitting is unnatural and not likely to lead to optimal estimates. On the other hand, one can deduce (13) from (12) by sending this symmetry to infinity; indeed, after applying a shift to (12) we see that
for any , and on sending
we obtain (13) (one could invoke the monotone convergence theorem here to justify the limit, though in this case it is simple enough that one can just use first principles).
Note that the principle of preserving symmetry only applies to direct approaches to proving inequalities such as (1). There is a complementary approach, discussed for instance in this previous post, which is to spend the symmetry to place the variable “without loss of generality” in a “normal form”, “convenient coordinate system”, or a “good gauge”. Abstractly: suppose that there is some subset
of
with the property that every
can be expressed in the form
for some
and
(that is to say,
). Then, if one wishes to prove an inequality (1) for all
, and one knows that both sides
of this inequality are
-invariant, then it suffices to check (1) just for those
in
, as this together with the
-invariance will imply the same inequality (1) for all
in
. By restricting to those
in
, one has given up (or spent) the
-invariance, as the set
will in typical not be preserved by the group action
. But by the same token, by eliminating the invariance, one also eliminates the prohibition on using non-invariant proof techniques, and one is now free to use a wider range of inequalities in order to try to establish (1). Of course, such inequalities should make crucial use of the restriction
, for if they did not, then the arguments would work in the more general setting
, and then the previous principle would again kick in and warn us that the use of non-invariant inequalities would be inefficient. Thus one should “spend” the symmetry wisely to “buy” a restriction
that will be of maximal utility in calculations (for instance by setting as many annoying factors and terms in one’s analysis to be
or
as possible).
As a simple example of this, let us revisit the complex triangle inequality (9). As already noted, both sides of this inequality are invariant with respect to the phase rotation symmetry . This seems to limit one to using phase-rotation-invariant techniques to establish the inequality, in particular ruling out the use of real and imaginary parts as discussed previously. However, we can instead spend the phase rotation symmetry to restrict to a special class of
and
. It turns out that the most efficient way to spend the symmetry is to achieve the normalisation of
being a nonnegative real; this is of course possible since any complex number
can be turned into a nonnegative real by multiplying by an appropriate phase
. Once
is a nonnegative real, the imaginary part disappears and we have
and the triangle inequality (9) is now an immediate consequence of (10), (11). (But note that if one had unwisely spent the symmetry to normalise, say, to be a non-negative real, then one is no closer to establishing (9) than before one had spent the symmetry.)
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