Compact Sets, Unit Balls, Norms, Inner Products: Delightful Reads

[qspeechc]

Hi everyone. I wasn't really sure where to put this thread so I stuck it here, which seems the closest fit.

Anyway I've been thinking about this for too long: what characterises the unit ball of a norm? Let's be specific: consider a finite-dimensional vector space, which may as well be $\mathbb{R}^n$.

Now I think that every convex and compact set C containing the origin in its interior is the unit ball of some norm. I suspect convexity is too strong a condition and we can get away with less, but I can't think what weaker conditions will work. Define all the points lying on the boundary of the set C to have norm one. Now for any vector x in the space there must be some real number $\lambda$ such that $\lambda x$ lies on the boundary of C. Then define the norm of x to be $|\lambda |^{-1}$, and the norm of the origin to be zero. I think I have a proof that such a $\lambda$ does indeed exist; uniqueness may be troublesome but should follow from the convexity (this is why I want convexity); at least it is intuitively clear that this number exists and is unique. Now let's check this is actually a norm we have defined:

1) It's clear the norm is always non-negative and zero only for the zero vector.

2) The scaling of vectors (positive homogeneity) is all dandy with respect to the norm.

3) The triangle inequality is tough. I have no idea why this should be true. Perhaps this follows from convexity too?

Am I vaguely on the right track? What exactly characterises the unit balls of norm in a Euclidean space? Now every inner product induces a norm, so what characterises unit balls arising from inner-products? How do they differ from a unit ball of a norm that doesn't come from an inner-product? Are all norms from some inner-product?

Already this matter brings up many questions, and there are even more if we generalise. What about if we think of a vector space of arbitrary dimension, not necessarily finite? I suspect we can say very little in this case, simply because almost none of the nice properties finite dimensional vector spaces have carries over to the infinite dimensional case, but I have nothing precise to say in this case. What about vector spaces over an arbitrary field? I suppose there are many other closely related questions, all very interesting, which I have not asked but I would like to know about as well.

Any help? I know I've asked a lot of questions, so can you point me toward the literature so I can read up about this? Thank-you.

 

[LCKurtz]

For any norm, you must have ||-x|| = ||x||, which implies that the unit ball must be symmetric through the origin. So, for example, even in R² your convex set and hence your unit ball can't be an equilateral triangle. Or any other triangle.

 

[qspeechc]

Yes, I recognised that minutes before I found your post. So then, is every compact set which is symmetric about the origin the unit ball of some norm? I dropped the convexity condition because it is too strong. The unit ball for the lp space is not convex for p<1:
http://en.wikipedia.org/wiki/Unit_sphere
I still haven't gotten any closer to answering these questions.

 

[LCKurtz]

But convexity is necessary. If x and y are on the unit ball,

$$||tx + (1-t)y||\le ||tx||+||(1-t)y||\le t+1-t = 1\hbox{ for }0\le t\le 1$$

If p < 1 the l^p spaces aren't normed.