Related papers: A Sharp Bound on Large Planar Signed Vector Sums

Let ||.|| be a norm in R^d whose unit ball is B. Assume that V\subset B is a finite set of cardinality n, with \sum_{v \in V} v=0. We show that for every integer k with 0 \le k \le n, there exists a subset U of V consisting of k elements…

Consider a set of $r$ convex $d$-polytopes $P_1,P_2,...,P_r$, where $d\ge{}3$ and $r\ge{}2$, and let $n_i$ be the number of vertices of $P_i$, $1\le{}i\le{}r$. It has been shown by Fukuda and Weibel that the number of $k$-faces of the…

An important theorem of Banaszczyk (Random Structures & Algorithms `98) states that for any sequence of vectors of $\ell_2$ norm at most $1/5$ and any convex body $K$ of Gaussian measure $1/2$ in $\mathbb{R}^n$, there exists a signed…

The Shapley-Folkman theorem shows that Minkowski averages of uniformly bounded sets tend to be convex when the number of terms in the sum becomes much larger than the ambient dimension. In optimization, Aubin and Ekeland [1976] show that…

A generalization of pairwise intersecting Minkowski arrangement of centrally symmetric convex bodies is the pairwise intersecting Minkowski arrangement of order $\mu$. Here, the homothetic copies of a centrally symmetric convex body are so…

We establish a sharp geometric constant for the upper bound on the resonance counting function for surfaces with hyperbolic ends. An arbitrary metric is allowed within some compact core, and the ends may be of hyperbolic planar, funnel, or…

In this paper, we deal with the successive inner and outer radii with respect to Orlicz Minkowski sum. The upper and lower bounds for the radii of the Orlicz Minkowski sum of two convex bodies are established.

Consider a sum $S_n=v_i\varepsilon_1+\cdots+v_n\varepsilon_{n}$, where $(v_i)^{n}_{i=1}$ are non-zero vectors in $\mathbb{R}^{d}$ and $(\varepsilon_i)^{n}_{i=1}$ are independent Rademacher random variables (i.e.,…

We study lower bounds for the Minkowski and Hausdorff dimensions of the algebraic sum E+K of two subsets E and K of d-dimensional Euclidean space.

For a general family of graphs on $\mathbb{Z}^n$, we translate the edge-isoperimetric problem into a continuous isoperimetric problem in $\mathbb{R}^n$. We then solve the continuous isoperimetric problem using the Brunn-Minkowski inequality…

We consider the problem of choosing Euclidean points to maximize the sum of their weighted pairwise distances, when each point is constrained to a ball centered at the origin. We derive a dual minimization problem and show strong duality…

We obtain bounds on the least dimension of an affine space that can contain an $n$-dimensional submanifold without any pairs of parallel or intersecting tangent lines at distinct points. This problem is closely related to the generalized…

A graph is called (generically) rigid in $\mathbb{R}^d$ if, for any choice of sufficiently generic edge lengths, it can be embedded in $\mathbb{R}^d$ in a finite number of distinct ways, modulo rigid transformations. Here we deal with the…

We consider a generalization of the hyperplane problem to arbitrary measures in place of volume and to sections of lower dimensions. We prove this generalization for unconditional convex bodies and for duals of bodies with bounded volume…

We algebraically compute all possible sectional curvature values for canonical algebraic curvature tensors, and use this result to give a method for constructing general sectional curvature bounds. We use a well-known method to…

The paper characterizes the convex hull of the closure of the cone-volume set $C_\cv(U)$, consisting of all cone-volume vectors of polygons with outer unit normals vectors contained in $U$, for any finite set $U \subseteq \R^2, \pos(U) =…

The study of (minimally) rigid graphs is motivated by numerous applications, mostly in robotics and bioinformatics. A major open problem concerns the number of embeddings of such graphs, up to rigid motions, in Euclidean space. We capture…

Let $A_1,A_2,...,A_n$ be the vertices of a polygon with unit perimeter, that is $\sum_{i=1}^n |A_i A_{i+1}|=1$. We derive various tight estimates on the minimum and maximum values of the sum of pairwise distances, and respectively sum of…

Define a boundary point of a graph which is embedded in the Euclidean plane a vertex which is incident to only one edge. In this paper we consider graphs which are embedded in the Euclidean plane with a finite number of boundary points. The…

The colorful simplicial depth of a collection of d+1 finite sets of points in Euclidean d-space is the number of choices of a point from each set such that the origin is contained in their convex hull. We use methods from combinatorial…