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


On 19 Mar 2008 13:15:00 -0400, israel@xxxxxxxxxxx wrote:



On Mar 18, 3:27 pm, Harald Hanche-Olsen <han...@xxxxxxxxxxxx> wrote:
+ docjack...@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.

Note that this hexagon has three distinct side-lengths. If the norm
was of the form of Conjecture 1, there would be three distinct linear
isometric involutions
(x_1 -> (+/-) x_1, x_2 -> (+/-) x_2). But it's obvious that
the only isometry of this hexagon other than the identity is
reflection through the origin.

I don't follow this at all - if we were talking about isometries
in the euclidean norm yes, but...

Hmm. Actually it seemed to me that Conjecture 1 was ambigulously
stated: "where x_i = <x,b_i>, i.e., x_i is the i-th component of x in
the basis mentioned". The "ie" is wrong, since those two statements
of what x_i should be are not the same. Maybe you're assuming
he really meant x_i = <x, b_i>?

Seems to me that he _should_ have meant that the x_i are the
coordinate functionals for the basis (b_1, b_2). In that case
there _is_ such a basis: Say B is the given polygon, and
define T(s,t) = (s, t - s/4). Then A = T(B) is a polygon with
vertices +-(0,1), +-(1,1/4) and +-(1,-1/4) (if I did the aritmetic
wrong adjust T so that A does have those vertices...)

So A is symmetric about both coordinate axes, so A is
the unit ball of a norm that depends only on the absolute
values of the coordinates. Hence B is the unit ball of a
norm that depends only on the absolute values of the
coordinate functionals wrt some basis.

Robert Israel israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada


David C. Ullrich
.

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