625 research outputs found

    On the Beer index of convexity and its variants

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    Let SS be a subset of Rd\mathbb{R}^d with finite positive Lebesgue measure. The Beer index of convexity b⁥(S)\operatorname{b}(S) of SS is the probability that two points of SS chosen uniformly independently at random see each other in SS. The convexity ratio c⁥(S)\operatorname{c}(S) of SS is the Lebesgue measure of the largest convex subset of SS divided by the Lebesgue measure of SS. We investigate the relationship between these two natural measures of convexity. We show that every set S⊆R2S\subseteq\mathbb{R}^2 with simply connected components satisfies b⁥(S)≀αc⁥(S)\operatorname{b}(S)\leq\alpha\operatorname{c}(S) for an absolute constant α\alpha, provided b⁥(S)\operatorname{b}(S) is defined. This implies an affirmative answer to the conjecture of Cabello et al. that this estimate holds for simple polygons. We also consider higher-order generalizations of b⁥(S)\operatorname{b}(S). For 1≀k≀d1\leq k\leq d, the kk-index of convexity b⁥k(S)\operatorname{b}_k(S) of a set S⊆RdS\subseteq\mathbb{R}^d is the probability that the convex hull of a (k+1)(k+1)-tuple of points chosen uniformly independently at random from SS is contained in SS. We show that for every d≄2d\geq 2 there is a constant ÎČ(d)>0\beta(d)>0 such that every set S⊆RdS\subseteq\mathbb{R}^d satisfies b⁥d(S)≀ÎČc⁥(S)\operatorname{b}_d(S)\leq\beta\operatorname{c}(S), provided b⁥d(S)\operatorname{b}_d(S) exists. We provide an almost matching lower bound by showing that there is a constant Îł(d)>0\gamma(d)>0 such that for every Δ∈(0,1)\varepsilon\in(0,1) there is a set S⊆RdS\subseteq\mathbb{R}^d of Lebesgue measure 11 satisfying c⁥(S)≀Δ\operatorname{c}(S)\leq\varepsilon and b⁥d(S)≄γΔlog⁥21/Δ≄γc⁥(S)log⁥21/c⁥(S)\operatorname{b}_d(S)\geq\gamma\frac{\varepsilon}{\log_2{1/\varepsilon}}\geq\gamma\frac{\operatorname{c}(S)}{\log_2{1/\operatorname{c}(S)}}.Comment: Final version, minor revisio

    The convexification effect of Minkowski summation

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    Let us define for a compact set A⊂RnA \subset \mathbb{R}^n the sequence A(k)={a1+⋯+akk:a1,
,ak∈A}=1k(A+⋯+A⏟k times). A(k) = \left\{\frac{a_1+\cdots +a_k}{k}: a_1, \ldots, a_k\in A\right\}=\frac{1}{k}\Big(\underset{k\ {\rm times}}{\underbrace{A + \cdots + A}}\Big). It was independently proved by Shapley, Folkman and Starr (1969) and by Emerson and Greenleaf (1969) that A(k)A(k) approaches the convex hull of AA in the Hausdorff distance induced by the Euclidean norm as kk goes to ∞\infty. We explore in this survey how exactly A(k)A(k) approaches the convex hull of AA, and more generally, how a Minkowski sum of possibly different compact sets approaches convexity, as measured by various indices of non-convexity. The non-convexity indices considered include the Hausdorff distance induced by any norm on Rn\mathbb{R}^n, the volume deficit (the difference of volumes), a non-convexity index introduced by Schneider (1975), and the effective standard deviation or inner radius. After first clarifying the interrelationships between these various indices of non-convexity, which were previously either unknown or scattered in the literature, we show that the volume deficit of A(k)A(k) does not monotonically decrease to 0 in dimension 12 or above, thus falsifying a conjecture of Bobkov et al. (2011), even though their conjecture is proved to be true in dimension 1 and for certain sets AA with special structure. On the other hand, Schneider's index possesses a strong monotonicity property along the sequence A(k)A(k), and both the Hausdorff distance and effective standard deviation are eventually monotone (once kk exceeds nn). Along the way, we obtain new inequalities for the volume of the Minkowski sum of compact sets, falsify a conjecture of Dyn and Farkhi (2004), demonstrate applications of our results to combinatorial discrepancy theory, and suggest some questions worthy of further investigation.Comment: 60 pages, 7 figures. v2: Title changed. v3: Added Section 7.2 resolving Dyn-Farkhi conjectur

    Statistical analysis of measures of non-convexity

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    Several measures of non-convexity (departures from convexity) have been introduced in the literature, both for sets and functions. Some of them are of geometric nature, while others are more of topological nature. We address the statistical analysis of some of these measures of non-convexity of a set SS, by dealing with their estimation based on a sample of points in SS. We introduce also a new measure of non-convexity. We discuss briefly about these different notions of non-convexity, prove consistency and find the asymptotic distribution for the proposed estimators. We also consider the practical implementation of these estimators and illustrate their applicability to a real data example

    Positive Definite ℓ1\ell_1 Penalized Estimation of Large Covariance Matrices

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    The thresholding covariance estimator has nice asymptotic properties for estimating sparse large covariance matrices, but it often has negative eigenvalues when used in real data analysis. To simultaneously achieve sparsity and positive definiteness, we develop a positive definite ℓ1\ell_1-penalized covariance estimator for estimating sparse large covariance matrices. An efficient alternating direction method is derived to solve the challenging optimization problem and its convergence properties are established. Under weak regularity conditions, non-asymptotic statistical theory is also established for the proposed estimator. The competitive finite-sample performance of our proposal is demonstrated by both simulation and real applications.Comment: accepted by JASA, August 201

    An Institutional Frame to Compare Alternative Market Designs in EU Electricity Balancing

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    The so-called Ăą electricity wholesale marketĂą is, in fact, a sequence of several markets. The chain is closed with a provision for Ăą balancing,Ăą in which energy from all wholesale markets is balanced under the authority of the Transmission Grid Manager (TSO in Europe, ISO in the United States). In selecting the market design, engineers in the European Union have traditionally preferred the technical role of balancing mechanisms as Ăą security mechanisms.Ăą They favour using penalties to restrict the use of balancing energy by market actors. While our paper in no way disputes the importance of grid security, nor the competency of engineers to elaborate the technical rules, we wish to attract attention to the real economic consequences of alternative balancing designs. We propose a numerical simulation in the framework of a two-stage equilibrium model. This simulation allows us to compare the economic properties of designs currently existing within the European Union and to measure their fallout. It reveals that balancing designs, which are typically presented as simple variants on technical security, are in actuality alternative institutional frameworks having at least four potential economic consequences: a distortion of the forward price; an asymmetric shift in the participantsĂą profits; an increase in the System OperatorĂą s revenues; and inefficiencies

    Smoothing ℓ1\ell_1-penalized estimators for high-dimensional time-course data

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    When a series of (related) linear models has to be estimated it is often appropriate to combine the different data-sets to construct more efficient estimators. We use ℓ1\ell_1-penalized estimators like the Lasso or the Adaptive Lasso which can simultaneously do parameter estimation and model selection. We show that for a time-course of high-dimensional linear models the convergence rates of the Lasso and of the Adaptive Lasso can be improved by combining the different time-points in a suitable way. Moreover, the Adaptive Lasso still enjoys oracle properties and consistent variable selection. The finite sample properties of the proposed methods are illustrated on simulated data and on a real problem of motif finding in DNA sequences.Comment: Published in at http://dx.doi.org/10.1214/07-EJS103 the Electronic Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of Mathematical Statistics (http://www.imstat.org

    Design of optimal corrective taxes in the alcohol market

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    The Topology of Wireless Communication

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    In this paper we study the topological properties of wireless communication maps and their usability in algorithmic design. We consider the SINR model, which compares the received power of a signal at a receiver against the sum of strengths of other interfering signals plus background noise. To describe the behavior of a multi-station network, we use the convenient representation of a \emph{reception map}. In the SINR model, the resulting \emph{SINR diagram} partitions the plane into reception zones, one per station, and the complementary region of the plane where no station can be heard. We consider the general case where transmission energies are arbitrary (or non-uniform). Under that setting, the reception zones are not necessarily convex or even connected. This poses the algorithmic challenge of designing efficient point location techniques as well as the theoretical challenge of understanding the geometry of SINR diagrams. We achieve several results in both directions. We establish a form of weaker convexity in the case where stations are aligned on a line. In addition, one of our key results concerns the behavior of a (d+1)(d+1)-dimensional map. Specifically, although the dd-dimensional map might be highly fractured, drawing the map in one dimension higher "heals" the zones, which become connected. In addition, as a step toward establishing a weaker form of convexity for the dd-dimensional map, we study the interference function and show that it satisfies the maximum principle. Finally, we turn to consider algorithmic applications, and propose a new variant of approximate point location.Comment: 64 pages, appeared in STOC'1

    Exchange rate and market power in import price

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    This study consists of three papers in the area of international market analysis, as listed in Chapter 1, 2, and 3. Each paper has its own issue and application, but the main theme behind these papers is to figure out interactions of international firms\u27 real decisions with respect to changes in financial variables or structure attributing to the firms\u27 behaviors. The papers focus especially on a risk-averse international firm\u27s decision model with respect to fluctuations in exchange rates;The first two papers relate the international firm\u27s ex-ante real decision to the portfolio theory in correspondence to recent importance of managing risk. Chapter 1 deals with interactions between diversification strategy and currency hedging by futures contracts when a competitive & risk-averse importing agent chooses optimal import quantities and hedging levels under dual uncertainties of price and exchange rate. The resulting total import level under the scheme depends significantly on the degree of correlation among relevant currencies; that is because the currency hedging virtually determines the covariance effect of portfolio variance. Chapter 2 introduces another risk-diversification model in determining the input mixture within a framework of the capital-asset-price-model. The Chinese wheat import market is empirically analyzed to justify this portfolio approach and to explain potential conflicts between the buyer\u27s risk diversification efforts and suppliers\u27 market power. While concentrating on the risk reduction effect, these papers support hedging roles of currency futures contracts among the advanced markets in Chapter 1 and of diversification strategy in importing non-homogenous products in Chapter 2;As an illustration of the market structure related to demand functions, Chapter 3 deals with the topic of pass-through in terms of the oligopoly pricing conduct in the market. To find out the nature of demand convexity, this study draws several testable implications and also evaluates an empirical example of the import beer pricing in the US. Given the open debate on the stability of the level of pass-through, a Kalman filter estimation is adapted in the empirical application
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