A Characterization Theorem and An Algorithm for A Convex Hull Problem

Given $S= \{v_1, \dots, v_n\} \subset \mathbb{R} ^m$ and $p \in \mathbb{R} ^m$, testing if $p \in conv(S)$, the convex hull of $S$, is a fundamental problem in computational geometry and linear programming. First, we prove a Euclidean {\it distance duality}, distinct from classical separation theorems such as Farkas Lemma: $p$ lies in $conv(S)$ if and only if for each $p' \in conv(S)$ there exists a {\it pivot}, $v_j \in S$ satisfying $d(p',v_j) \geq d(p,v_j)$. Equivalently, $p \not \in conv(S)$ if and only if there exists a {\it witness}, $p' \in conv(S)$ whose Voronoi cell relative to $p$ contains $S$. A witness separates $p$ from $conv(S)$ and approximate $d(p, conv(S))$ to within a factor of two. Next, we describe the {\it Triangle Algorithm}: given $\epsilon \in (0,1)$, an {\it iterate}, $p' \in conv(S)$, and $v \in S$, if $d(p, p') < \epsilon d(p,v)$, it stops. Otherwise, if there exists a pivot $v_j$, it replace $v$ with $v_j$ and $p'$ with the projection of $p$ onto the line $p'v_j$. Repeating this process, the algorithm terminates in $O(mn \min \{\epsilon^{-2}, c^{-1}\ln \epsilon^{-1} \})$ arithmetic operations, where $c$ is the {\it visibility factor}, a constant satisfying $c \geq \epsilon^2$ and $\sin (\angle pp'v_j) \leq 1/\sqrt{1+c}$, over all iterates $p'$. Additionally, (i) we prove a {\it strict distance duality} and a related minimax theorem, resulting in more effective pivots; (ii) describe $O(mn \ln \epsilon^{-1})$-time algorithms that may compute a witness or a good approximate solution; (iii) prove {\it generalized distance duality} and describe a corresponding generalized Triangle Algorithm; (iv) prove a {\it sensitivity theorem} to analyze the complexity of solving LP feasibility via the Triangle Algorithm. The Triangle Algorithm is practical and competitive with the simplex method, sparse greedy approximation and first-order methods.Comment: 42 pages, 17 figures, 2 tables. This revision only corrects minor typo

From optimal transportation to optimal teleportation

The object of this paper is to study estimates of $\epsilon^{-q}W_p(\mu+\epsilon\nu, \mu)$ for small $\epsilon>0$. Here $W_p$ is the Wasserstein metric on positive measures, $p>1$, $\mu$ is a probability measure and $\nu$ a signed, neutral measure ($\int d\nu=0$). In [W1] we proved uniform (in $\epsilon$) estimates for $q=1$ provided $\int \phi d\nu$ can be controlled in terms of the $\int|\nabla\phi|^{p/(p-1)}d\mu$, for any smooth function $\phi$. In this paper we extend the results to the case where such a control fails. This is the case where if, e.g. $\mu$ has a disconnected support, or if the dimension of $\mu$ , $d$ (to be defined) is larger or equal $p/(p-1)$. In the later case we get such an estimate provided $1/p+1/d\not=1$ for $q=\min(1, 1/p+1/d)$. If $1/p+1/d=1$ we get a log-Lipschitz estimate. As an application we obtain H\"{o}lder estimates in $W_p$ for curves of probability measures which are absolutely continuous in the total variation norm . In case the support of $\mu$ is disconnected (corresponding to $d=\infty$) we obtain sharp estimates for $q=1/p$ ("optimal teleportation"): $$\lim_{\epsilon\rightarrow 0}\epsilon^{-1/p}W_p(\mu, \mu+\epsilon\nu) = \|\nu\|_{\mu}$$ where $\|\nu\|_{\mu}$ is expressed in terms of optimal transport on a metric graph, determined only by the relative distances between the connected components of the support of $\mu$, and the weights of the measure $\nu$ in each connected component of this support.Comment: 24 pages, 3 figure

Spin splitting and Kondo effect in quantum dots coupled to noncollinear ferromagnetic leads

We study the Kondo effect in a quantum dot coupled to two noncollinear ferromagnetic leads. First, we study the spin splitting $\delta\epsilon=\epsilon_{\downarrow}-\epsilon_{\uparrow}$ of an energy level in the quantum dot by tunnel couplings to the ferromagnetic leads, using the Poor man's scaling method. The spin splitting takes place in an intermediate direction between magnetic moments in the two leads. $\delta\epsilon \propto p\sqrt{\cos^2(\theta/2)+v^2\sin^2(\theta/2)}$, where $p$ is the spin polarization in the leads, $\theta$ is the angle between the magnetic moments, and $v$ is an asymmetric factor of tunnel barriers ($-1<v<1$). Hence the spin splitting is always maximal in the parallel alignment of two ferromagnets ($\theta=0$) and minimal in the antiparallel alignment ($\theta=\pi$). Second, we calculate the Kondo temperature $T_{\mathrm{K}}$. The scaling calculation yields an analytical expression of $T_{\mathrm{K}}$ as a function of $\theta$ and $p$, $T_{\mathrm{K}}(\theta, p)$, when $\delta\epsilon \ll T_{\mathrm{K}}$. $T_{\mathrm{K}}(\theta, p)$ is a decreasing function with respect to $p\sqrt{\cos^2(\theta/2)+v^2\sin^2(\theta/2)}$. When $\delta\epsilon$ is relevant, we evaluate $T_{\mathrm{K}}(\delta\epsilon, \theta, p)$ using the slave-boson mean-field theory. The Kondo resonance is split into two by finite $\delta\epsilon$, which results in the spin accumulation in the quantum dot and suppression of the Kondo effect.Comment: 11 pages, 8 figures, revised versio

Maximal operators and differentiation theorems for sparse sets

We study maximal averages associated with singular measures on \rr. Our main result is a construction of singular Cantor-type measures supported on sets of Hausdorff dimension $1 - \epsilon$, $0 \leq \epsilon < {1/3}$ for which the corresponding maximal operators are bounded on $L^p(\mathbb R)$ for $p > (1 + \epsilon)/(1 - \epsilon)$. As a consequence, we are able to answer a question of Aversa and Preiss on density and differentiation theorems in one dimension. Our proof combines probabilistic techniques with the methods developed in multidimensional Euclidean harmonic analysis, in particular there are strong similarities to Bourgain's proof of the circular maximal theorem in two dimensions. Updates: Andreas Seeger has provided an argument to the effect that our global maximal operators are in fact bounded on L^p(R) for all p>1; in particular, it follows that our differentiation theorems are also valid for all p>1. Furthermore, David Preiss has proved that no such differentiation theorems (let alone maximal estimates) can hold with p=1. These arguments are included in the new version. We have also improved the exposition in a number of places.Comment: Revised version. The final version will appear in Duke Math.

Geometry of Rounding

Rounding has proven to be a fundamental tool in theoretical computer science. By observing that rounding and partitioning of $\mathbb{R}^d$ are equivalent, we introduce the following natural partition problem which we call the {\em secluded hypercube partition problem}: Given $k\in \mathbb{N}$ (ideally small) and $\epsilon>0$ (ideally large), is there a partition of $\mathbb{R}^d$ with unit hypercubes such that for every point $p \in \mathbb{R}^d$, its closed $\epsilon$-neighborhood (in the $\ell_{\infty}$ norm) intersects at most $k$ hypercubes? We undertake a comprehensive study of this partition problem. We prove that for every $d\in \mathbb{N}$, there is an explicit (and efficiently computable) hypercube partition of $\mathbb{R}^d$ with $k = d+1$ and $\epsilon = \frac{1}{2d}$. We complement this construction by proving that the value of $k=d+1$ is the best possible (for any $\epsilon$) for a broad class of ``reasonable'' partitions including hypercube partitions. We also investigate the optimality of the parameter $\epsilon$ and prove that any partition in this broad class that has $k=d+1$, must have $\epsilon\leq\frac{1}{2\sqrt{d}}$. These bounds imply limitations of certain deterministic rounding schemes existing in the literature. Furthermore, this general bound is based on the currently known lower bounds for the dissection number of the cube, and improvements to this bound will yield improvements to our bounds. While our work is motivated by the desire to understand rounding algorithms, one of our main conceptual contributions is the introduction of the {\em secluded hypercube partition problem}, which fits well with a long history of investigations by mathematicians on various hypercube partitions/tilings of Euclidean space