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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 aExpand
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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 someExpand
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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.
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 $$\text {Gr}^{\ge 0}(n,k)$$Expand
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Optimality of codes with respect to error probability in Gaussian noise
TLDR
We consider geometrical optimization problems related to optimizing the error probability in the presence of a Gaussian noise, in particular, the conjecture about minimizing of the Gaussian measure of a simplex. Expand
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On the Circle Covering Theorem by A.W. Goodman and R.E. Goodman
TLDR
In 1945, A.W. Goodman proved the following conjecture by P.E. Erdős: Given a family of (round) disks of radii r_1, r_n in the plane, it is always possible to cover them by a disk of radius $$R = \sum r_i$$R=∑ri, provided they cannot be separated into two subfamilies by a straight line disjoint from the disks. Expand
Waist of maps measured via Urysohn width
We discuss various questions of the following kind: for a continuous map $X \to Y$ from a compact metric space to a simplicial complex, can one guarantee the existence of a fiber large in the senseExpand
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 forExpand
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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 cylinderExpand
Dependence of the heavily covered point on parameters.
We examine Gromov's method of selecting a point "heavily covered" by simplices formed by a given finite point sets, in order to understand the dependence of the heavily covered point on parameters.Expand