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Results tagged with convex-geometry
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user 2554
A branch of geometry dealing with convex sets and functions. Polytopes, convex bodies, discrete geometry, linear programming, antimatroids, ...
7
votes
Approaching convex and discrete geometry from other disciplines
For convex geometry & functional analysis:
Ball, Keith Convex geometry and functional analysis. Handbook of the geometry of Banach spaces, Vol. I, 161–194, North-Holland, Amsterdam, 2001.
Ball, Ke …
- 31.5k
1
vote
A lower bound on $\|\cdot\|_{p_{\ast}}$ image of $\ell^{q_{\ast}}$ vectors
This is really a comment rather than an answer, but it is too long for a comment and I feel compelled to say something because you used my tag. :)
I do not understand the first statement of your pro …
- 31.5k
4
votes
Accepted
A lower bound on $\|\cdot\|_{p_{\ast}}$ image of $\ell^{q_{\ast}}$ vectors
Not being able to sleep due to jet lag, I had a chance to think about your problem a bit. The answer is elementary and simple, but I will say more than is necessary.
First, people interested in quan …
- 31.5k
4
votes
Finite dimensional subspaces of $L^1.$
There are many characterizations of Banach spaces that embed isometrically into $L_1$. See
Wells, J. H.; Williams, L. R. Embeddings and extensions in analysis. Ergebnisse der Mathematik und ihrer Gr …
- 31.5k
5
votes
2
votes
Accepted
Closure of the interior of a convex set in a topological vector space
Yes. Assume WLOG that $0$ is an interior point and $y$ is in $C$. Draw a picture to see that the interval $[0,y)$ is contained in the interior of $C$. Specifically, if $U$ is a neighborhood of $0$ c …
- 31.5k
9
votes
Do random projections (approximately) preserve convexity?
Negative result:
See p. 377 in Chapter 15 of Matousek's book, which can be found
here. In short, if you want the image of the $k$ points to be between the surface of a convex body $K$ and the surf …
- 31.5k
9
votes
Intuitive explanation of Dvoretzky's theorem
There is a more difficult proof than the quantitative finite dimensional proofs that gives only the qualitative version of Dvoretzky's theorem but is arguably more intuitive. You use Ramsey's theorem …
- 31.5k
3
votes
Characterization of $l_p$ up to a linear isometry
Not an answer, but too long for a comment:
Look at Wells and Williams book, Embeddings and extensions in analysis, Springer Ergebnisse 84. There you find the classical description of what spaces $X$ …
- 31.5k
4
votes
Maximal Ellipsoid
Ben, John's theorem is easier for unit balls of spaces which have a one symmetric basis because you can argue that the basis vectors must be orthogonal with respect to the Euclidean norm determined by …
- 31.5k
8
votes
Continuous automorphism groups of normed vector spaces?
I think what groups can be the isometry group of a finite dimensional normed space are classified, maybe by Y. Gordon and/or D.R. Lewis. I don't have access to emath from home but will check the refer …
- 31.5k