Reverse and dual Loomis-Whitney-type inequalities

Various results are proved giving lower bounds for the $m$th intrinsic volume $V_m(K)$, $m=1,\dots,n-1$, of a compact convex set $K$ in ${\mathbb{R}}^n$, in terms of the $m$th intrinsic volumes of its projections on the coordinate hyperplanes (or its intersections with the coordinate hyperplanes). The bounds are sharp when $m=1$ and $m=n-1$. These are reverse (or dual, respectively) forms of the Loomis-Whitney inequality and versions of it that apply to intrinsic volumes. For the intrinsic volume $V_1(K)$, which corresponds to mean width, the inequality obtained confirms a conjecture of Betke and McMullen made in 1983

Convergence in shape of Steiner symmetrizations

There are sequences of directions such that, given any compact set K in R^n, the sequence of iterated Steiner symmetrals of K in these directions converges to a ball. However examples show that Steiner symmetrization along a sequence of directions whose differences are square summable does not generally converge. (Note that this may happen even with sequences of directions which are dense in S^{n-1}.) Here we show that such sequences converge in shape. The limit need not be an ellipsoid or even a convex set. We also deal with uniformly distributed sequences of directions, and with a recent result of Klain on Steiner symmetrization along sequences chosen from a finite set of directions.Comment: 11 page

Technical and Economic Overview of the Cellulignin Production Process

The paper analyses the chemical conversion process of biomass based on a diluted acid hydrolysis step, the Biomass Energy Process (BEM). The discussion starts from the technical point of view on the base of information reported literature, experimental laboratory analysis and pilot plant tests and then the authors tentatively explore the energy related costs of the process giving a comparison of the biomass derived solid fuel and fossil fuels. An evaluation of the product (cellulignin powder compacted into briquettes) and considered as a fuel and other energy source from ossil resources like oil, natural gas and coal, is reported considering the energy density and energy cost. The main technical feature of the process is the conversion of almost every type of bug wood biomass with large range of chemical and physics characteristics. The hydrolysis step determines the energy price and the threshold cost of the cellulignin is calculated at 60-70 euro/t about over which the NG energy is preferable. Environmental concerns can be however considered

Intersections of Dilatates of Convex Bodies

We initiate a systematic investigation into the nature of the function ∝K(L,ρ) that gives the volume of the intersection of one convex body K in Rn and a dilatate ρL of another convex body L in Rn, as well as the function ηK(L, ρ) that gives the (n - 1)-dimensional Hausdorff measure of the intersection of K and the boundary ∂(ρ L) of ρL. The focus is on the concavity properties of αK (L, ρ). Of particular interest is the case when K and L are symmetric with respect to the origin. In this situation, there is an interesting change in the concavity properties of αK (L, ρ) between dimension 2 and dimensions 3 or higher. When L is the unit ball, an important special case with connections to E. Lutwak\u27s dual Brunn-Minkowski theory, we prove that this change occurs between dimension 2 and dimensions 4 or higher, and conjecture that it occurs between dimension 3 and dimension 4. We also establish an isoperimetric inequality with equality condition for subsets of equatorial zones in the sphere S2, and apply this and the Brunn-Minkowski inequality in the sphere to obtain results related to this conjecture, as well as to the properties of a new type of symmetral of a convex body, which we call the equatorial symmetral

Convergence of symmetrization processes

Steiner and Schwarz symmetrizations, and their most important relatives, the Minkowski, Minkowski-Blaschke, fiber, inner rotational, and outer rotational symmetrizations, are investigated. The focus is on the convergence of successive symmetrals with respect to a sequence of $i$-dimensional subspaces of $\mathbb{R}^n$. Such a sequence is called universal for a family of sets if the successive symmetrals of any set in the family converge to a ball with center at the origin. New universal sequences for the main symmetrizations, for all valid dimensions $i$ of the subspaces, are found by combining two groups of new results. In the first, a theorem of Klain for Steiner symmetrization is extended to Schwarz, Minkowski, Minkowski-Blaschke, and fiber symmetrizations, showing that if a sequence of subspaces is drawn from a finite set ${\mathcal{F}}$ of subspaces, the successive symmetrals of any compact convex set converge to a compact convex set that is symmetric with respect to any subspace in ${\mathcal{F}}$ appearing infinitely often in the sequence. The second group of results provides finite sets ${\mathcal{F}}$ of subspaces such that symmetry with respect to each subspace in ${\mathcal{F}}$ implies full rotational symmetry. It is also proved that for Steiner, Schwarz, and Minkowski symmetrizations, a sequence of $i$-dimensional subspaces is universal for the class of compact sets if and only if it is universal for the class of compact convex sets, and Klain's theorem is shown to hold for Schwarz symmetrization of compact sets

Performance of electro-osmotic dewatering on different types of sewage sludge

The feasibility of pressure-driven electro-dewatering (EDW) on sludge samples taken after different biological processes, stabilisation methods or mechanical dewatering techniques was assessed. First, the influence of potential values on EDW of anaerobically and aerobically stabilised, mechanically dewatered, sludge samples was investigated. Preliminary tests carried out by applying a constant potential (10, 15 and 20 V) in a lab-scale device confirmed the possibility to reach a dry solid (DS) content of up to 42.9%, which corresponds to an increase of 15% of the dry content in dewatered sludge without the application of the electrical field. Dewatering increased with the applied potential but at the expense of a higher energy consumption. A potential equal to 15 V was chosen as the best compromise for EDW performance, in terms of DS content and energy consumption. Then, the influence of the mechanical dewatering was studied on aerobically stabilised sludge samples with a lower initial DS content: the higher initial water content led to a lower final DS content but with a considerable reduction of energy consumption. Finally, the biological process, studied by comparing sludge samples from conventional activated sludge and membrane bioreactor processes, didn’t evidence any influence on EDW. Experimental results shown that DS obtained after mechanical dewatering, volatile solids and conductivity are the main factors influencing EDW. Anaerobically digested sludge reached the highest DS content, thanks to lower organic fraction