Re: Characterization of the norms in finite dimensional real vector spaces


On 18 Mar 2008 18:27:07 -0400, Harald Hanche-Olsen
<hanche@xxxxxxxxxxxx> wrote:



+ docjack666@xxxxxxxxx:

On 18 mar, 06:15, Harald Hanche-Olsen <han...@xxxxxxxxxxxx> wrote:
The standard characterization is geometrical: Let B be the closed unit
ball for any given norm on R^n. Then B is a compact balanced convex set
with 0 in its interior, and moreover any such B is the closed unit ball
for a norm. (By balanced I mean: if x belongs to B then so does -x.)

Thank you very much. I had some trouble finding a proof of this
characterization. Could you please tell me where I can find it?

Not sure if it's merely folklore, but I would guess that many
books on functional analysis would contain this: See the proof of the
Hahn-Banach separation theorem, in which the construction of a
subadditive functional from a convex set with 0 in the interior is an
important ingredient. Getting from there to the norm is an easy
exercise.

I think you can see from this characterization that your two
conjectures are way too strong, even for n=2.

Conjecture 2 was a long shot, but I'm still inclined to believe
conjecture 1 can be proven... Is it clear to you it's false? Could you
please point out why?

In R^2, consider the polygon with corners at ±(1,0), ±(0,1) and
±(1,1/2). I am pretty sure (but am too lazy to prove rigorously) that
this set is assymmetric enough to fit your conjecture 1.

Taking n = 2:

If a norm has unit ball B then it satisfies his condition if and only
if there exists an invertible linear map T such that A = T(B)
is symmetric about both coordinate axes.

Say B is a polygon with exactly six vertices a, b, c, -a, -b, -c.
It's enough to find T such that T(a) = (0,1), T(b)_1 = T(c)_1
and T(b)_2 = -T(c)_2. That leads to four equations in four
variables - I wouldn't be surprised if there was always a solution
(maybe not).

I bet there's a counterexample consisting of a polygon with
8 or 10 vertices.

Of course we're having all the fun here...
I was also wondering... maybe one can prove that given a balanced
convex compact set B with 0 in its interior, then the border of B is
homotopic to the standard (euclidean) unit sphere... Any ideas?

Just map one to the other via radial lines.

David C. Ullrich
.

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