625 research outputs found
On the Beer index of convexity and its variants
Let be a subset of with finite positive Lebesgue measure.
The Beer index of convexity of is the probability
that two points of chosen uniformly independently at random see each other
in . The convexity ratio of is the Lebesgue
measure of the largest convex subset of divided by the Lebesgue measure of
. We investigate the relationship between these two natural measures of
convexity.
We show that every set with simply connected
components satisfies for an
absolute constant , provided 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 . For
, the -index of convexity of a set
is the probability that the convex hull of a
-tuple of points chosen uniformly independently at random from is
contained in . We show that for every there is a constant
such that every set satisfies
, provided
exists. We provide an almost matching lower bound by
showing that there is a constant such that for every
there is a set of Lebesgue
measure satisfying and
.Comment: Final version, minor revisio
The convexification effect of Minkowski summation
Let us define for a compact set the sequence It was independently proved by Shapley, Folkman and Starr (1969)
and by Emerson and Greenleaf (1969) that approaches the convex hull of
in the Hausdorff distance induced by the Euclidean norm as goes to
. We explore in this survey how exactly approaches the convex
hull of , 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 , 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 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
with special structure. On the other hand, Schneider's index possesses a strong
monotonicity property along the sequence , and both the Hausdorff
distance and effective standard deviation are eventually monotone (once
exceeds ). 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
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 ,
by dealing with their estimation based on a sample of points in . 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 Penalized Estimation of Large Covariance Matrices
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 -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
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 -penalized estimators for high-dimensional time-course data
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 -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
The Topology of Wireless Communication
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
-dimensional map. Specifically, although the -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 -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
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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