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Results tagged with convex-geometry
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user 21123
A branch of geometry dealing with convex sets and functions. Polytopes, convex bodies, discrete geometry, linear programming, antimatroids, ...
10
votes
Accepted
Reference request: Ehrhart's conjecture on the geometry of numbers
I guess it must be in bad taste to answer my own question, but I now know a bit more about this problem and maybe there are other people interested in references to it.
As I mentioned in the question …
- 13.5k
9
votes
Research trends in geometry of numbers?
I can't say that what I'll relate is fundamental, but it does fit into the new ideas category. Since I and (my collaborator) Florent Balacheff have given talks on the subject and the paper will be in …
- 13.5k
8
votes
Accepted
A question about a question about 3-dimensional convex bodies
I think this is true. A proof would go like this:
First prove that the body must be an ellipsoid.
Without loss of generality you may assume that your body $B$ contains the origin as an interior poin …
- 13.5k
5
votes
Quermassintegrals as mean curvature integrals
These things hold in amazing generality, but you must dig into bit of geometric measure theory, which is the right language for this. Here is the paper that probably started the industry in this dire …
- 13.5k
4
votes
Accepted
Minkowski successive minima inequality for a lattice base?
Yes, a version of Minkowski's successive minima studied by Mahler and Weyl consists in letting $\lambda_i'$ to be the radius of the smallest ball containing $i$ linearly independent lattice vectors t …
- 13.5k
4
votes
Isometric (?) embedding problem.
Dear Igor,
Since the answer to your problem is "no" I'm not going to dwell on that, but I feel there is something interesting behind your question that does not admit such a clear cut answer.
Two k …
- 13.5k
4
votes
Set of Positive Definite matrices with determinant > 1 forms a convex set
I'm not sure if you are asking for the proof of the theorem assuming Minkowski's inequality or if you are asking for the proof including a proof of Minkowski's inequality.
If you assume Minkowski's …
- 13.5k
3
votes
A conjecture of parallelogram inside convex and central symmetric curve
This is an old result in convex geometry. See
E. Sas, ¨Uber ein Extremumeigenschaft der Ellipsen, Compositio Math. 6 (1939) 468–
470.
A. M. Macbeath, An extremal property of the hypersphere, Proc. C …
- 13.5k
3
votes
Polar body of a convex body that avoids a lattice
This is not really an answer, but a comment on a natural variation of the original problem: assume a convex body in $\mathbb{R}^n$ that is not necessarily centrally symmetric contains only one lattice …
- 13.5k
1
vote
Asymmetry of projections
This answer is a summary of discussions with Dmitry Ryabogin and Ralph Howard. The conclusion is that the original problem is equivalent to other long-standing open problems in convex geometry.
OP. L …
- 13.5k