A conjecture about Lp tunings

Assume that V is the space of monzos and that T is some temperament.

Then the conjecture is this: for any Lp norm on V and temperament T,
there exists a projection map P where rank(P) = rank(T) such that P(m)
= min ||T^-1(T(m))|| for m in V.

In other words, there exists some projection map P sending any monzo m
to the unique monzo which is in the same coset as m, but which has
minimum norm. This would generalize the TE and Frobenius projection
maps to arbitrary Lp projection maps.

Another way to phrase the above conjecture is: for any Lp norm on V
and temperament T, there is at least one subspace S in V where rank(S)
= rank(T) such that ||s|| = ||T^-1(T(s))|| for all s in S.

Any ideas?

-Mike

On Fri, Jul 20, 2012 at 11:52 PM, Mike Battaglia <battaglia01@gmail.com>
wrote:
>
> Another way to phrase the above conjecture is: for any Lp norm on V
> and temperament T, there is at least one subspace S in V where rank(S)
> = rank(T) such that ||s|| = ||T^-1(T(s))|| for all s in S.

Yeah, screw that plan. Doesn't work. I just Monte Carlo'd myself some
different test cases. This only ends up working out for L2, where you
get an actual subspace. For other norms you get all sorts of weird
algebraic varieties instead. So there goes that plan.

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
For other norms you get all sorts of weird
> algebraic varieties instead. So there goes that plan.

Cool! What kind of weird algebraic varieties?

On Sun, Jul 22, 2012 at 12:13 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...>
> wrote:
> For other norms you get all sorts of weird
> > algebraic varieties instead. So there goes that plan.
>
> Cool! What kind of weird algebraic varieties?

For instance, for L3 in R^3, you get some weird looking manifold. I'm
not sure how to post it here though, since unless you can rotate it
it's hard to see. If I post a list of the vectors approximately lying
on the manifold, will you be able to plot it in Maple?

I think the basic idea is that to find the vector in the coset with
the min L2 norm, you need to differentiate and find where the
derivative is 0. If you're differentiating, then all of those square
terms in the L2 norm summation become linear terms, and consequently
the result is a linear subspace. It's the same reason least squares
problems are so easy to solve. For L3, on the other hand, when you
differentiate they become quadratic terms, and the resulting locus of
points minimizing the L3 norm ends up having some quadratic curve to
it. Or something.

-Mike "just a hunch" Battaglia

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> > Cool! What kind of weird algebraic varieties?
>
> For instance, for L3 in R^3, you get some weird looking manifold. I'm
> not sure how to post it here though, since unless you can rotate it
> it's hard to see. If I post a list of the vectors approximately lying
> on the manifold, will you be able to plot it in Maple?

I was interested in its algebraic properties, but it was idle curiosity.