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 $s$-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 $A$ of $C_b(K)$ is dense if and only if the set of all strong peak points is a norming subset of $A$. 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 $X$ is a norming subset of $\mathcal{P}({}^n X)$, then the set of all strongly norm attaining elements in $\mathcal{P}({}^n X)$ is dense. In particular, the set of all points at which the norm of $\mathcal{P}({}^n X)$ is Fr\'echet differentiable is a dense $G_\delta$ 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 $X$ with an absolute norm, its polynomial numerical indices are one if and only if $X$ is isometric to $\ell_\infty^n$. 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