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Category Archives: Convexity
Yair Shenfeld and Ramon van Handel Settled (for polytopes) the Equality Cases For The Alexandrov-Fenchel Inequalities
Two weeks ago, I participated (remotely) in the discrete geometry Oberwolfach meeting, and Ramon van Handel gave a beautiful lecture about the equality cases of Alexandrov-Fenchel inequalities which is among the most famous problems in convex geometry. In the top … Continue reading
Posted in Combinatorics, Convexity, Geometry
Tagged Igor Pak, Ramon van Handel, Rolf Schneider, Swee Hong Chan, Yair Shenfeld
2 Comments
On the Limit of the Linear Programming Bound for Codes and Packing
Alex Samorodnitsky The most powerful general method for proving upper bounds for the size of error correcting codes and of spherical codes (and sphere packing) is the linear programming method that goes back to Philippe Delsarte. There are very interesting … Continue reading
Posted in Combinatorics, Convexity, Geometry
Tagged Alex Samorodnitsky, error-correcting codes, Philippe Delsarte, spherical codes
2 Comments
Marcelo Campos, Matthew Jenssen, Marcus Michelen and, and Julian Sahasrabudhe: Striking new Lower Bounds for Sphere Packing in High Dimensions
A few days ago, a new striking paper appeared on the arXiv A new lower bound for sphere packing by Marcelo Campos, Matthew Jenssen, Marcus Michelen, and Julian Sahasrabudhe Here is the abstract: We show there exists a packing of … Continue reading
On Viazovska’s modular form inequalities by Dan Romik
The main purpose of this post is to tell you about a recent paper by Dan Romik which gives a direct proof of two crucial inequalities in Maryna Viazovska’s proof that lattice sphere packing is the densest sphere packing in … Continue reading
Posted in Combinatorics, Convexity, Geometry, Number theory
Tagged Dan Romik, Henry Cohn, Maryna Viazovska, Noam Elkies
4 Comments
Progress Around Borsuk’s Problem
I was excited to see the following 5-page paper: Convex bodies of constant width with exponential illumination number by Andrii Arman, Andrii Bondarenko, and Andriy Prymak Abstract: We show that there exist convex bodies of constant width in with illumination … Continue reading
What is the maximum number of Tverberg’s partitions?
The problem presented in this post was discussed in my recent lecture “New types of order types” in the workshop on discrete convexity and geometry in Budapest, a few weeks ago. The lecture described various results and questions including the … Continue reading
A High-Dimensional Diameter Problem for Polytopes
Avi Wigderson is here for a year and it was a good opportunity to go back together to the question of diameter of polytopes. The diameter problem for polytopes is to determine the behavior of the maximum diameter of the … Continue reading
Posted in Combinatorics, Convex polytopes, Convexity, Polymath3
Tagged diameter, high-dimensional combinatorics, Hirsch conjecture, Polymath3
5 Comments
Bo’az Klartag and Joseph Lehec: The Slice Conjecture Up to Polylogarithmic Factor!
Bo’az Klartag (right) and Joseph Lehec (left) In December 2020, we reported on Yuansi Chen breakthrough result on Bourgain’s alicing problem and the Kannan Lovasz Simonovits conjecture. It is a pleasure to report on a further fantastic progress on these … Continue reading
Posted in Analysis, Computer Science and Optimization, Convexity, Geometry, Probability
Tagged Bo'az Klartag, Joseph Lehec
4 Comments
Test Your intuition 51
Suppose that and are two compact convex sets in space. Suppose that contains . Now consider two quantities is the average volume of a simplex forms by four points in drawn uniformly at random. is the average volume of a … Continue reading
Posted in Convexity, Geometry, Probability, Test your intuition
Tagged Test your intuition
12 Comments
Combinatorics and more 