204 research outputs found
The geometry of p-convex intersection bodies
Busemann's theorem states that the intersection body of an origin-symmetric
convex body is also convex. In this paper we provide a version of Busemann's
theorem for p-convex bodies. We show that the intersection body of a p-convex
body is q-convex for certain q. Furthermore, we discuss the sharpness of the
previous result by constructing an appropriate example. This example is also
used to show that IK, the intersection body of K, can be much farther away from
the Euclidean ball than K. Finally, we extend these theorems to some general
measure spaces with log-concave and -concave measure
Local minimality of the volume-product at the simplex
It is proved that the simplex is a strict local minimum for the volume
product, P(K)=min(vol(K) vol(K^z)), K^z is the polar body of K with respect to
z, the minimum is taken over z in the interior of K, in the Banach-Mazur space
of n-dimensional (classes of ) convex bodies. Linear local stability in the
neighborhood of the simplex is proved as well. The proof consists of an
extension to the non-symmetric setting of methods that were recently introduced
by Nazarov, Petrov, Ryabogin and Zvavitch, as well as proving results of
independent interest, concerning stability of square order of volumes of polars
of non-symmetric convex bodies.Comment: Mathematika, accepte
Strong peak points and strongly norm attaining points with applications to denseness and polynomial numerical indices
Using the variational method, it is shown that the set of all strong peak
functions in a closed algebra of is dense if and only if the set
of all strong peak points is a norming subset of . As a corollary we can
induce the denseness of strong peak functions on other certain spaces. In case
that a set of uniformly strongly exposed points of a Banach space is a
norming subset of , then the set of all strongly norm
attaining elements in is dense. In particular, the set of
all points at which the norm of is Fr\'echet
differentiable is a dense subset.
In the last part, using Reisner's graph theoretic-approach, we construct some
strongly norm attaining polynomials on a CL-space with an absolute norm. Then
we show that for a finite dimensional complex Banach space with an absolute
norm, its polynomial numerical indices are one if and only if is isometric
to . Moreover, we give a characterization of the set of all
complex extreme points of the unit ball of a CL-space with an absolute norm
Measurement of PM2.5 Mass Concentration Using an Electrostatic Particle Concentrator-Based Quartz Crystal Microbalance
Particulate matter (PM) is one of the most critical air pollutants, and various instruments have been developed to measure PM mass concentration. Of these, quartz crystal microbalance (QCM) based instruments have received much attention. However, these instruments are subject to significant drawbacks: particle bounce due to poor adhesion, need for frequent cleanings of the crystal electrode, and non-uniform distribution of collected particles. In this study, we present an electrostatic particle concentrator (EPC)-based QCM (qEPC) instrument capable of measuring the mass concentration of PM 2.5 (PM smaller than 2.5 ??m), while avoiding the drawbacks. Experimental measurements showed high collection efficiencies (~99% at 1.2 liters/min), highly uniform particle distributions for long sampling periods (up to 120 min at 50 ??g/m 3 ), and high mass concentration sensitivity [0.068(Hz/min)/(??g/m 3 )]. The enhanced uniformity of particle deposition profiles and mass concentration sensitivity were made possible by the unique flow and electrical design of the qEPC instrument
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