Related papers: A `converse' to the Constraint Lemma

The convex hulls of face-vertex incident vectors of 3-face-colorable convex polytopes are computed. It is found that every such convex hull is a $d$-polytope with $d+2$ or $d+3$ vertices. Utilizing Gale transform and Gale diagram, we…

We develop a topological approach to prove the generalized Lax conjecture using the fact that determinants of sufficiently big symmetric linear pencils are able to express the rigidly convex sets of RZ polynomials of any degree $d$.…

We give improved lower bounds for binary $3$-query locally correctable codes (3-LCCs) $C \colon \{0,1\}^k \rightarrow \{0,1\}^n$. Specifically, we prove: (1) If $C$ is a linear design 3-LCC, then $n \geq 2^{(1 - o(1))\sqrt{k} }$. A design…

We characterize the three-dimensional spaces admitting at least six or at least seven equidistant points. In particular, we show the existence of $C^\infty$ norms on $\R^3$ admitting six equidistant points, which refutes a conjecture of…

In this paper, we prove the following result conjectured by Z.-W. Sun: $$ (2n-1){3n\choose n}| \sum_{k=0}^{n}{6k\choose 3k}{3k\choose k}{6(n-k)\choose 3(n-k)}{3(n-k)\choose n-k}. $$ by showing that the left-hand side divides each summand on…

This paper analyzes the convex hull of the parametric curve $\left(t, t^2, \cdots, t^N\right)$, where $t$ is in a closed interval. It finds that every point in the convex hull is representable as a convex combination of $\frac{N+1}{2}$…

The main goal of this paper is to provide an alternative proof of the following theorem of Petty: in the normed space of dimension at least three, every 3-element equilateral set can be extended to a 4-element equilateral set. Our approach…

Let $L$ be a set of $n$ lines in $\reals^d$, for $d\ge 3$. A {\em joint} of $L$ is a point incident to at least $d$ lines of $L$, not all in a common hyperplane. Using a very simple algebraic proof technique, we show that the maximum…

In this note we study a conjecture by Jer\'onimo-Castro, Magazinov and Sober\'on which generalized a question posed by Dol'nikov. Let $F_1,F_2,\dots,F_n$ be families of translates of a convex compact set $K$ in the plane so that each two…

We call a CNF formula linear if any two clauses have at most one variable in common. Let m(k) be the largest integer m such that any linear k-CNF formula with <= m clauses is satisfiable. We show that 4^k / (4e^2k^3) <= m(k) < ln(2) k^4…

For a set $A$ of points in the plane, not all collinear, we denote by ${\rm tr}(A)$ the number of triangles in any triangulation of $A$; that is, ${\rm tr}(A) = 2i+b-2$ where $b$ and $i$ are the numbers of points of $A$ in the boundary and…

Problem 4.19 in Ziegler's "Lectures on Polytopes" asserts that every simple $3$-dimensional polytope has the property that its dual can be constructed as the convex hull of a subset of the vertices of the original simple polytope. In this…

We consider bounded integer knapsacks where the weights and variable upper bounds together form a superincreasing sequence. The elements of this superincreasing knapsack are exactly those vectors that are lexicographically smaller than the…

For an integer \( k \geq 2 \), the sequence of \( k \)-generalized Lucas numbers is defined by the recurrence relation \( L_n^{(k)} = L_{n-1}^{(k)} + \cdots + L_{n-k}^{(k)} \) for all \( n \geq 2 \), with initial conditions \( L_0^{(k)} = 2…

In this paper we show a a proof by explicit bijections of the famous Kirkman-Cayley formula for the number of dissections of a convex polygon. Our starting point is the bijective correspondence between the set of nested sets made by \(k\)…

We study the occurrence of curved three-point configurations in fractal subsets of the real line. We prove that if \(E \subset [0,1]\) is a compact set with sufficiently large Hausdorff dimension, then \(E\) contains a curved three-point…

We give a proof to the following theorem, which is well-known among experts: A connected subcomplex $W$ of a finite dimensional CAT(0) cubed complex $X$ is convex if and only if Lk$(v, W)$ is a full subcomplex of Lk$(v, X)$ for every vertex…

A \emph{thrackle} is a graph drawn in the plane so that every pair of its edges meet exactly once, either at a common end vertex or in a proper crossing. Conway's thrackle conjecture states that the number of edges is at most the number of…

The union-closed sets conjecture, also known as Frankl's conjecture, is a well-studied problem with various formulations. In terms of lattices, the conjecture states that every finite lattice $L$ with more than one element contains a…

This article focuses on the occurrence of 3-point configurations in subsets of $\mathbb{R}^d$ of sufficient thickness. We prove that a compact set $A\subset \mathbb{R}^d$ contains a similar copy of any linear $3$-point configuration (such…