Dual of the Weil norm proof
(Archived from Facebook, originally a post by Mike Battaglia addressing Gene Smith)
OK, so I proved the conjecture. The dual to the weil norm is max(<V 0|) - min(<V 0|). I'll formulate monzos as column vectors and vals as row vectors below.
Let M be the space of p-limit real monzos with the Weil norm on them. The Weil norm of a weighted monzo |a b c ...> is 1/2*(|a| + |b| + |c| + ... + |a+b+c+...|), giving you log(max(n,d)).
Note that this still looks like an L1 norm - in fact, it's the L1 norm of the "augmented monzo"
1/2 * |a b c ... ; (a+b+c+...)>
where we have appended the sum of the coefficients to the end of the vector as a new coefficient.
We can left-multiply our original monzos by a matrix to transform them into these special augmented monzos. That transformation matrix just so happens to be
[.5 0 0 ...]
[0 .5 0 ...]
[0 0 .5 ...]
[... ... ... ...]
[.5 .5 .5 ...]
which is a diagonal matrix with all .5's on the diagonal, followed by an extra row at the bottom of all .5's. I will call this matrix Ξ. So if you take any monzo m, L1(Ξm) = Weil(m).
Since I called the original monzo space M, I will call this new, larger space M°; it is the L1 normed space of p-limit augmented monzos that have one extra coordinate tacked on the end. Ξ can be thought of as taking monzos in M to a subspace of M° - the subspace where the last coordinate is the sum of the others. Thus, you can see that the Weil norm itself can be thought of as the restriction of the L1 norm on M° to this subspace. By an abuse of notation, I will also call this subspace M.
Now, let's see if we can use this to figure out what the dual norm on vals looks like.
Let's say that the dual space to M is called V, and the dual space to M° is called V°. V is the space of real p-limit vals; V° is the space of p-limit augmented vals with one extra coordinate tacked on at the end.
We know that the norm on V° is the Linf norm, since the norm on M° is the L1 norm. Our goal is to use this to figure out the norm on V.
Let's say that ~M is the subspace of V° which annhilates M; e.g. it consists of all augmented vals which map M to 0. By a useful corollary of the Hahn-Banach theorem (https://web.archive.org/web/20201130153651/https://math.unl.edu/~s-bbockel1/928/node25.html), the space V is isometrically isomorphic to the quotient space V°/~M with the quotient norm on it. We will use this to compute an expression for the norm on V, aka the dual to the Weil norm.
Since M is the column space of Ξ, then ~M is the left nullspace of Ξ. It happens to be the 1D subspace of V° spanned by <1 1 1 ...; -1|. I will call this augmented val J°, since it's the JIP with this augmented coordinate -1 at the end.
A good way to make sense of this interesting situation is to note that Ξ also acts as a matrix taking augmented vals to normal ones. So, we can find the norm of an arbitrary val v in V as follows: look for all augmented vals in the preimage Ξ^-1(v) and find the one with the shortest Linf norm. The value of this Linf norm is the norm on v. We'll see if we can find a nice closed-form expression for this.
To do so, first we note that for any val v = <x y z ...|, the augmented val v° = <2x 2y 2z ...; 0| with 0 tacked on at the end is in its preimage. So the norm on v is given by minimizing the expression
Linf(v° + rj°)
for r in R. Or, in other words, we want to minimize
Linf(<2x 2y 2z ...; 0| + <r r r...; -r|)
We can rewrite the above as follows
inf_r max(|2x+r|, |2y+r|, |2z+r|, ..., |r|)
OK, so what's the r that minimizes that? It turns out to be the r such that
max(2x+r,2y+r,...,r) = -min(2x+r,2y+r,...,r)
If r diverges from that in either direction, either the max coordinate increases or the min coordinate decreases, both of which have the effect of increasing the absolute value of that coordinate and hence the overall max.
Put differently, we want to take the range of <2x 2y 2z ...; 0| and set r so that this range is centered around zero.
But now we're done. We don't need to worry about finding r. If the range is centered around zero, then max(abs(···)) over all coordinates is going to be half the range, and the above proves that this is going to be the minimized-max that we were searching for.
So finally, we prove that the quotient norm on v° is half its range, e.g. (max(v°)-min(v°))/2. And since we ended up doubling all of the coefficients of v when we translated it to v°, the division by 2 cancels that out, proving the original conjecture that ||v||* = max(<v 0|) - min(<v 0|)