Related papers: Optimal Vector Balancing for Zonotopes

We investigate the convex-body containment problem $\max\{s >0 : s Z \subseteq Q\}$, where the outer body $Q \subseteq \mathbb R^d$ is described by a membership oracle and the inner body $Z \subseteq \mathbb R^d$ is a zonotope. Our main…

We say that two sets $S,T\subset\{1,2,\dots,n\}$ are chord separated if there does not exist a cyclically ordered quadruple $a,b,c,d$ of integers satisfying $a,c\in S-T$ and $b,d\in T-S$. This is a weaker version of Leclerc and Zelevinsky's…

Let $G_1,\dots,G_m$ be independent copies of the standard gaussian random vector in $\mathbb{R}^d$. We show that there is an absolute constant $c$ such that for any $A \subset S^{d-1}$, with probability at least $1-2\exp(-c\Delta m)$, for…

We prove the following generalisation of Schauder's fixed point conjecture: Let $C_1,...,C_n$ be convex subsets of a Hausdorff topological vector space. Suppose that the $C_i$ are closed in $C=C_1\cup...\cup C_n$. If $f:C\to C$ is a…

Based on the concepts of a generalized critical point and the corresponding generalized P.S. condition introduced by Duong Minh Duc[1], we have proved a new $Z_2$ index theorem and get a result on multiplicity of generalized critical…

We find a new regular solution of six-dimensional Einstein's equations with a positive cosmological constant. It has the same isometry group as the (deformed) conifold geometry, and the superpotential approach is used to solve the equations…

We consider the question of determining the structure of the set of all $d$-dimensional vectors of the form $N^{-1}(1_A*1_{-A}(x_1), ..., 1_A*1_{-A}(x_d))$ for $A \subseteq \{1,...,N\}$, and also the set of all $(2N+1)^{-1}(1_B*1_B(x_1),…

We find classically stable solitons (instantons) in odd (even) dimensional scalar noncommutative field theories whose scalar potential, $V(\ph)$, has at least two minima. These solutions are bubbles of the false vacuum whose size is set by…

The constant $C_A(n)$ is defined to be the smallest natural number $k$ such that any sequence of $k$ elements in $\mathbb Z_n$ has a subsequence of consecutive terms whose $A$-weighted sum is zero, where the weight set $A\subseteq \mathbb…

A conic bundle is a contraction $X\to Z$ between normal varieties of relative dimension $1$ such that $-K_X$ is relatively ample. We prove a conjecture of Shokurov which predicts that, if $X\to Z$ is a conic bundle such that $X$ has…

Let $p$ be a homogeneous polynomial of degree $n$ in $n$ variables, $p(z_1,...,z_n) = p(Z)$, $Z \in C^{n}$. We call such a polynomial $p$ {\bf H-Stable} if $p(z_1,...,z_n) \neq 0$ provided the real parts $Re(z_i) > 0, 1 \leq i \leq n$. This…

We prove that for any finite real hyperplane arrangement the average projection volumes of the maximal cones is given by the coefficients of the characteristic polynomial of the arrangement. This settles the conjecture of Drton and Klivans…

Let $X^k_{m,n}=\Sigma^k (\mathbb R\mathbb P^m/\mathbb R\mathbb P^n)$. In this note we completely determine the values of $k,m,n$ for which the total Stiefel-Whitney class $w(\xi)=1$ for any vector bundle $\xi$ over $X^k_{m,n}$.

Generalizing Milnor's result that an FTC (finite total curvature) knot has an isotopic inscribed polygon, we show that any two nearby knotted FTC graphs are isotopic by a small isotopy. We also show how to obtain sharper constants when the…

Let $K$ be a unit ball of some norm in $R^n$. For an arbitrary direction $u\in R^n$, there is associated a unit-ball $K_u$, which is rotationally invariant with respect to rotations keeping $u$ fixed, called the $u$-spin of $K_u$. It is…

In this paper, I propose a new framework for representing multidimensional incomplete preferences through zonotope-valued utilities, addressing the shortcomings of traditional scalar and vector-based models in decision theory. Traditional…

In a finite dimensional Hilbert space, each normalized vector (state) can be chosen as a member of an orthonormal basis of the space. We give a proof of this statement in a manner that seems to be more comprehensible for physics students…

We prove that the uniform probability measure $\mu$ on every $(n-k)$-dimensional projection of the $n$-dimensional unit cube verifies the variance conjecture with an absolute constant $C$ $$\textrm{Var}_\mu|x|^2\leq C \sup_{\theta\in…

It is proven that a conjecture of Tao (2010) holds true for log-concave random variables on the integers: For every $n \geq 1$, if $X_1,\ldots,X_n$ are i.i.d. integer-valued, log-concave random variables, then $$ H(X_1+\cdots+X_{n+1}) \geq…

There is a positive constant $c_1$ such that for any diagram $D$ representing the unknot, there is a sequence of at most $2^{c_1 n}$ Reidemeister moves that will convert it to a trivial knot diagram, $n$ is the number of crossings in $D$. A…