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This is to get some clarity, I know the two words have nothing to do with each other but they sound so similar that I caught myself saying "locally convex" when I really meant "uniformly convex" and the other way around. Perhaps it would help me if you could tell me whether my understanding of why they are called that is correct and correct me if it's wrong. So here goes:

A uniformly convex space is a space such that for any two points $x,y$ that I pick on the unit ball, if the line between them is $\varepsilon > 0$ long then the midpoint $\frac{x+y}{2}$ is $\delta > 0$ away from the boundary of the ball. (by that I mean that the the absolute value of the midpoint is less than $1$)

I am not sure how this is much different from saying that the unit ball is convex. It makes me think that every point in the space has a convex neighbourhood basis that is, that for every neighbourhood $N$ I can find a ball inside it (by scaling and shifting of the unit ball) that is contained in $N$. Not sure I am using the right words. This understanding must be wrong because if that was the case one should call the space "locally convex". Perhaps this is the cause of my confusion?

On the other hand, as far as I understand, a locally convex space is a vector space together with a collection of seminorms that is endowed with the weakest topology such that all the seminorms are continuous. I think this means that any neighbourhood of any point in the space contains an intersection of open balls $B_{\|\cdot\|_\alpha}$ where $\|\cdot\|_\alpha$ are seminorms. So here it seems more obvious that I should call the space locally convex, since if I pick a neighbourhood I can intersect convex open sets to get a convex open set inside it.

So my question is: which parts of my current understanding are right and which aren't? Thanks for your help.

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    Yes, your suspicion is correct and holds independently of the dimension (as long as it is at least two, possibly infinite): you get uniform convexity for $1 \lt p \lt \infty$ (the reflexive range) and no uniform convexity for $p = 1,\infty$ (the unit sphere contains straight lines!).2012-08-15

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