12
$\begingroup$

I have a Banach space geometry question (a curiosity-driven spin-off from a research topic). Given a point $x$ on the unit sphere of a Banach space and a vector $y\ne 0$, there is a multiple $t_0y$ of $y$ for which $\|t_0y-x\|$ is minimized (this will be unique if the norm is strictly convex).

My question is this:

For which Banach spaces $X$ is it guaranteed that $\|t_0y\|\le \|x\|$?

My "Euclidean intuition" suggested that this should be the case for all Banach spaces, but a little experimentation showed that this is not the case. You quickly see this is really a question about two dimensions. In fact it seems to fail for every $\ell^p$, $p\ne 2$ (see the attached figure in $p=1.2$).

Could it be true that this property characterizes Hilbert space? (I looked at the obvious sources: (MO 11192 and papers mentioned in there and didn't find anything of the sort).

$\endgroup$

2 Answers 2

14
$\begingroup$

The answer is no in dimension 2 and yes in dimension 3 and higher. The property that the nearest-point projection to a line does not increase the norm is equivalent to the symmetry of orthogonality relation defined as follows: $x$ is orthogonal to $y$ iff $\|x+ty\|\ge\|x\|$ for all $t\in\mathbb R$.

It is well-known that symmetry of this orthogonality relation in dimension $\ge 3$ implies that the norm is Euclidean, see e.g. Thompson's "Minkowski geometry", Theorem 3.4.10.

This is not the case in dimension 2. There are many counter-examples (I believe they are called Radon planes). Basically you only need to ensure that every unit vector with its unit orthogonal one span a constant parallelogram area, this is easy to satisfy and is equivalent to the symmetry of orthogonality. For a simple explicit example (although non-smooth), consider a norm on the plane whose unit ball is a regular hexagon.

$\endgroup$
0
1
$\begingroup$

I do not know the answer, but I would suggest having a look at the book "Characterizations of inner product spaces" by Dan Amir, which features, who would have guessed, many many characterizations of inner product spaces. There is also a book called "Inner product structures" by Istratescu which I have found useful.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.