The Alexandrov--Fenchel inequality bounds from below the square of the mixed volume $V(K_1,K_2,K_3,ldots,K_n)$ of convex bodies $K_1,ldots,K_n$ in $mathbb{R}^n$ by the product of the mixed volumes $V(K_1,K_1,K_3,ldots,K_n)$ and $V(K_2,K_2,K_3,ldots,K
_n)$. As a consequence, for integers $alpha_1,ldots,alpha_minmathbb{N}$ with $alpha_1+cdots+alpha_m=n$ the product $V_n(K_1)^{frac{alpha_1}{n}}cdots V_n(K_m)^{frac{alpha_m}{n}} $ of suitable powers of the volumes $V_n(K_i)$ of the convex bodies $K_i$, $i=1,ldots,m$, is a lower bound for the mixed volume $V(K_1[alpha_1],ldots,K_m[alpha_m])$, where $alpha_i$ is the multiplicity with which $K_i$ appears in the mixed volume. It has been conjectured by Ulrich Betke and Wolfgang Weil that there is a reverse inequality, that is, a sharp upper bound for the mixed volume $V(K_1[alpha_1],ldots,K_m[alpha_m])$ in terms of the product of the intrinsic volumes $V_{alpha_i}(K_i)$, for $i=1,ldots,m$. The case where $m=2$, $alpha_1=1$, $alpha_2=n-1$ has recently been settled by the present authors (2020). The case where $m=3$, $alpha_1=alpha_2=1$, $alpha_3=n-2$ has been treated by Artstein-Avidan, Florentin, Ostrover (2014) under the assumption that $K_2$ is a zonoid and $K_3$ is the Euclidean unit ball. The case where $alpha_2=cdots=alpha_m=1$, $K_1$ is the unit ball and $K_2,ldots,K_m$ are zonoids has been considered by Hug, Schneider (2011). Here we substantially generalize these previous contributions, in cases where most of the bodies are zonoids, and thus we provide further evidence supporting the conjectured reverse Alexandrov--Fenchel inequality. The equality cases in all considered inequalities are characterized. More generally, stronger stability results are established as well.
The upgraded LHCb detector, due to start datataking in 2022, will have to process an average data rate of 4~TB/s in real time. Because LHCbs physics objectives require that the full detector information for every LHC bunch crossing is read out and ma
de available for real-time processing, this bandwidth challenge is equivalent to that of the ATLAS and CMS HL-LHC software read-out, but deliverable five years earlier. Over the past six years, the LHCb collaboration has undertaken a bottom-up rewrite of its software infrastructure, pattern recognition, and selection algorithms to make them better able to efficiently exploit modern highly parallel computing architectures. We review the impact of this reoptimization on the energy efficiency of the real-time processing software and hardware which will be used for the upgrade of the LHCb detector. We also review the impact of the decision to adopt a hybrid computing architecture consisting of GPUs and CPUs for the real-time part of LHCbs future data processing. We discuss the implications of these results on how LHCbs real-time power requirements may evolve in the future, particularly in the context of a planned second upgrade of the detector.
In stochastic geometry there are several instances of threshold phenomena in high dimensions: the behavior of a limit of some expectation changes abruptly when some parameter passes through a critical value. This note continues the investigation of t
he expected face numbers of polyhedral random cones, when the dimension of the ambient space increases to infinity. In the focus are the critical values of the observed threshold phenomena, as well as threshold phenomena for differences instead of quotients.
We consider an even probability distribution on the $d$-dimensional Euclidean space with the property that it assigns measure zero to any hyperplane through the origin. Given $N$ independent random vectors with this distribution, under the condition
that they do not positively span the whole space, the positive hull of these vectors is a random polyhedral cone (and its intersection with the unit sphere is a random spherical polytope). It was first studied by Cover and Efron. We consider the expected face numbers of these random cones and describe a threshold phenomenon when the dimension $d$ and the number $N$ of random vectors tend to infinity. In a similar way, we treat the solid angle, and more generally the Grassmann angles. We further consider the expected numbers of $k$-faces and of Grassmann angles of index $d-k$ when also $k$ tends to infinity.
Real-time data processing is one of the central processes of particle physics experiments which require large computing resources. The LHCb (Large Hadron Collider beauty) experiment will be upgraded to cope with a particle bunch collision rate of 30
million times per second, producing $10^9$ particles/s. 40 Tbits/s need to be processed in real-time to make filtering decisions to store data. This poses a computing challenge that requires exploration of modern hardware and software solutions. We present Compass, a particle tracking algorithm and a parallel raw input decoding optimised for GPUs. It is designed for highly parallel architectures, data-oriented and optimised for fast and localised data access. Our algorithm is configurable, and we explore the trade-off in computing and physics performance of various configurations. A CPU implementation that delivers the same physics performance as our GPU implementation is presented. We discuss the achieved physics performance and validate it with Monte Carlo simulated data. We show a computing performance analysis comparing consumer and server grade GPUs, and a CPU. We show the feasibility of using a full GPU decoding and particle tracking algorithm for high-throughput particle trajectories reconstruction, where our algorithm improves the throughput up to 7.4$times$ compared to the LHCb baseline.
Barthe proved that the regular simplex maximizes the mean width of convex bodies whose John ellipsoid (maximal volume ellipsoid contained in the body) is the Euclidean unit ball; or equivalently, the regular simplex maximizes the $ell$-norm of convex
bodies whose Lowner ellipsoid (minimal volume ellipsoid containing the body) is the Euclidean unit ball. Schmuckenschlager verified the reverse statement; namely, the regular simplex minimizes the mean width of convex bodies whose Lowner ellipsoid is the Euclidean unit ball. Here we prove stronger stabili
Croftons formula of integral geometry evaluates the total motion invariant measure of the set of $k$-dimensional planes having nonempty intersection with a given convex body. This note deals with motion invariant measures on sets of pairs of hyperpla
nes or lines meeting a convex body. Particularly simple results are obtained if, and only if, the given body is of constant width in the first case, and of constant brightness in the second case.
Poisson processes in the space of $(d-1)$-dimensional totally geodesic subspaces (hyperplanes) in a $d$-dimensional hyperbolic space of constant curvature $-1$ are studied. The $k$-dimensional Hausdorff measure of their $k$-skeleton is considered. Ex
plicit formulas for first- and second-order quantities restricted to bounded observation windows are obtained. The central limit problem for the $k$-dimensional Hausdorff measure of the $k$-skeleton is approached in two different set-ups: (i) for a fixed window and growing intensities, and (ii) for fixed intensity and growing spherical windows. While in case (i) the central limit theorem is valid for all $dgeq 2$, it is shown that in case (ii) the central limit theorem holds for $din{2,3}$ and fails if $dgeq 4$ and $k=d-1$ or if $dgeq 7$ and for general $k$. Also rates of convergence are studied and multivariate central limit theorems are obtained. Moreover, the situation in which the intensity and the spherical window are growing simultaneously is discussed. In the background are the Malliavin-Stein method for normal approximation and the combinatorial moment structure of Poisson U-statistics as well as tools from hyperbolic integral geometry.
The concept of splitting tessellations and splitting tessellation processes in spherical spaces of dimension $dgeq 2$ is introduced. Expectations, variances and covariances of spherical curvature measures induced by a splitting tessellation are studi
ed using tools from spherical integral geometry. Also the spherical pair-correlation function of the $(d-1)$-dimensional Hausdorff measure is computed explicitly and compared to its analogue for Poisson great hypersphere tessellations. Finally, the typical cell distribution and the distribution of the typical spherical maximal face of any dimension $kin{1,ldots,d-1}$ are expressed as mixtures of the related distributions of Poisson great hypersphere tessellations. This in turn is used to determine the expected length and the precise birth time distribution of the typical maximal spherical segment of a splitting tessellation.
