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Equality cases in Viterbo's conjecture and isoperimetric billiard inequalities
In this note we apply the billiard technique to deduce some results on Viterbo's conjectured inequality between volume of a convex body and its symplectic capacity. We show that the product of a
Elementary approach to closed billiard trajectories in asymmetric normed spaces
We apply the technique of K\'aroly Bezdek and Daniel Bezdek to study billiard trajectories in convex bodies, when the length is measured with a (possibly asymmetric) norm. We prove a lower bound for
A multi-plank generalization of the Bang and Kadets inequalities
If a convex body in $\mathbb{R}^n$ is covered by the union of convex bodies, multiple subadditivity questions can be asked. The subadditivity of the width is the subject of the celebrated plank
On the Circle Covering Theorem by A.W. Goodman and R.E. Goodman
It is shown that essentially the same idea may work for different analogues and generalizations of their result, and the following is proved: Given a family of positive homothetic copies of a fixed convex body, it is always possible to cover them by a translate of d+12(∑τi)K, provided they cannot be separated into two subfamilies by a hyperplane disjoint from the homothets.
On the Circle Covering Theorem by A .
In 1945, A.W. Goodman and R.E. Goodman proved the following conjecture by P. Erdős: Given a family of (round) disks of radii r1, . . ., rn in the plane, it is always possible to cover them by a disk
Flip cycles in plabic graphs
Planar bicolored (plabic) graphs are combinatorial objects introduced by Postnikov to give parameterizations of the positroid cells of the totally nonnegative Grassmannian $$\text {Gr}^{\ge 0}(n,k)$$
Another Ham Sandwich in the Plane
We show that every two nice measures in the plane can be partitioned into equal halves by translation of an angle from any k-fan when k is odd and in some cases when k is even. We also give some
Equality cases in Viterbo's conjecture related to permutohedra
In this note we show, using the billiard technique, that the product of a regular permutohedron and a regular simplex delivers an equality in Viterbo's conjecture.
Optimality of codes with respect to error probability in Gaussian noise
This work considers geometrical optimization problems related to optimizing the error probability in the presence of a Gaussian noise, and state related conjectures about the Gaussian measure, in particular, the conjecture about minimizing of theGaussian measure of a simplex.
Geometric Complexity of Planar Drawings
We say that a planar drawing of a graph is 1-thick if the distance between the images of any two vertices, a vertex and an edge, and two non-adjacent edges is at least 1. We prove that the cylinder