skip to main content


The NSF Public Access Repository (NSF-PAR) system and access will be unavailable from 5:00 PM ET until 11:00 PM ET on Friday, June 21 due to maintenance. We apologize for the inconvenience.

Title: Spherical convex hull of random points on a wedge

Consider two half-spaces$$H_1^+$$H1+and$$H_2^+$$H2+in$${\mathbb {R}}^{d+1}$$Rd+1whose bounding hyperplanes$$H_1$$H1and$$H_2$$H2are orthogonal and pass through the origin. The intersection$${\mathbb {S}}_{2,+}^d:={\mathbb {S}}^d\cap H_1^+\cap H_2^+$$S2,+d:=SdH1+H2+is a spherical convex subset of thed-dimensional unit sphere$${\mathbb {S}}^d$$Sd, which contains a great subsphere of dimension$$d-2$$d-2and is called a spherical wedge. Choosenindependent random points uniformly at random on$${\mathbb {S}}_{2,+}^d$$S2,+dand consider the expected facet number of the spherical convex hull of these points. It is shown that, up to terms of lower order, this expectation grows like a constant multiple of$$\log n$$logn. A similar behaviour is obtained for the expected facet number of a homogeneous Poisson point process on$${\mathbb {S}}_{2,+}^d$$S2,+d. The result is compared to the corresponding behaviour of classical Euclidean random polytopes and of spherical random polytopes on a half-sphere.

more » « less
Author(s) / Creator(s):
; ; ; ; ;
Publisher / Repository:
Springer Science + Business Media
Date Published:
Journal Name:
Mathematische Annalen
Medium: X Size: p. 2289-2316
["p. 2289-2316"]
Sponsoring Org:
National Science Foundation
More Like this
  1. Abstract

    Approximate integer programming is the following: For a given convex body$$K \subseteq {\mathbb {R}}^n$$KRn, either determine whether$$K \cap {\mathbb {Z}}^n$$KZnis empty, or find an integer point in the convex body$$2\cdot (K - c) +c$$2·(K-c)+cwhich isK, scaled by 2 from its center of gravityc. Approximate integer programming can be solved in time$$2^{O(n)}$$2O(n)while the fastest known methods for exact integer programming run in time$$2^{O(n)} \cdot n^n$$2O(n)·nn. So far, there are no efficient methods for integer programming known that are based on approximate integer programming. Our main contribution are two such methods, each yielding novel complexity results. First, we show that an integer point$$x^* \in (K \cap {\mathbb {Z}}^n)$$x(KZn)can be found in time$$2^{O(n)}$$2O(n), provided that theremaindersof each component$$x_i^* \mod \ell $$ximodfor some arbitrarily fixed$$\ell \ge 5(n+1)$$5(n+1)of$$x^*$$xare given. The algorithm is based on acutting-plane technique, iteratively halving the volume of the feasible set. The cutting planes are determined via approximate integer programming. Enumeration of the possible remainders gives a$$2^{O(n)}n^n$$2O(n)nnalgorithm for general integer programming. This matches the current best bound of an algorithm by Dadush (Integer programming, lattice algorithms, and deterministic, vol. Estimation. Georgia Institute of Technology, Atlanta, 2012) that is considerably more involved. Our algorithm also relies on a newasymmetric approximate Carathéodory theoremthat might be of interest on its own. Our second method concerns integer programming problems in equation-standard form$$Ax = b, 0 \le x \le u, \, x \in {\mathbb {Z}}^n$$Ax=b,0xu,xZn. Such a problem can be reduced to the solution of$$\prod _i O(\log u_i +1)$$iO(logui+1)approximate integer programming problems. This implies, for example thatknapsackorsubset-sumproblems withpolynomial variable range$$0 \le x_i \le p(n)$$0xip(n)can be solved in time$$(\log n)^{O(n)}$$(logn)O(n). For these problems, the best running time so far was$$n^n \cdot 2^{O(n)}$$nn·2O(n).

    more » « less
  2. Abstract

    Let$$(h_I)$$(hI)denote the standard Haar system on [0, 1], indexed by$$I\in \mathcal {D}$$ID, the set of dyadic intervals and$$h_I\otimes h_J$$hIhJdenote the tensor product$$(s,t)\mapsto h_I(s) h_J(t)$$(s,t)hI(s)hJ(t),$$I,J\in \mathcal {D}$$I,JD. We consider a class of two-parameter function spaces which are completions of the linear span$$\mathcal {V}(\delta ^2)$$V(δ2)of$$h_I\otimes h_J$$hIhJ,$$I,J\in \mathcal {D}$$I,JD. This class contains all the spaces of the formX(Y), whereXandYare either the Lebesgue spaces$$L^p[0,1]$$Lp[0,1]or the Hardy spaces$$H^p[0,1]$$Hp[0,1],$$1\le p < \infty $$1p<. We say that$$D:X(Y)\rightarrow X(Y)$$D:X(Y)X(Y)is a Haar multiplier if$$D(h_I\otimes h_J) = d_{I,J} h_I\otimes h_J$$D(hIhJ)=dI,JhIhJ, where$$d_{I,J}\in \mathbb {R}$$dI,JR, and ask which more elementary operators factor throughD. A decisive role is played by theCapon projection$$\mathcal {C}:\mathcal {V}(\delta ^2)\rightarrow \mathcal {V}(\delta ^2)$$C:V(δ2)V(δ2)given by$$\mathcal {C} h_I\otimes h_J = h_I\otimes h_J$$ChIhJ=hIhJif$$|I|\le |J|$$|I||J|, and$$\mathcal {C} h_I\otimes h_J = 0$$ChIhJ=0if$$|I| > |J|$$|I|>|J|, as our main result highlights: Given any bounded Haar multiplier$$D:X(Y)\rightarrow X(Y)$$D:X(Y)X(Y), there exist$$\lambda ,\mu \in \mathbb {R}$$λ,μRsuch that$$\begin{aligned} \lambda \mathcal {C} + \mu ({{\,\textrm{Id}\,}}-\mathcal {C})\text { approximately 1-projectionally factors through }D, \end{aligned}$$λC+μ(Id-C)approximately 1-projectionally factors throughD,i.e., for all$$\eta > 0$$η>0, there exist bounded operatorsABso thatABis the identity operator$${{\,\textrm{Id}\,}}$$Id,$$\Vert A\Vert \cdot \Vert B\Vert = 1$$A·B=1and$$\Vert \lambda \mathcal {C} + \mu ({{\,\textrm{Id}\,}}-\mathcal {C}) - ADB\Vert < \eta $$λC+μ(Id-C)-ADB<η. Additionally, if$$\mathcal {C}$$Cis unbounded onX(Y), then$$\lambda = \mu $$λ=μand then$${{\,\textrm{Id}\,}}$$Ideither factors throughDor$${{\,\textrm{Id}\,}}-D$$Id-D.

    more » « less
  3. Abstract

    The repeating fast radio burst FRB 20190520B is localized to a galaxy atz= 0.241, much closer than expected given its dispersion measure DM = 1205 ± 4 pc cm−3. Here we assess implications of the large DM and scattering observed from FRB 20190520B for the host galaxy’s plasma properties. A sample of 75 bursts detected with the Five-hundred-meter Aperture Spherical radio Telescope shows scattering on two scales: a mean temporal delayτ(1.41 GHz) = 10.9 ± 1.5 ms, which is attributed to the host galaxy, and a mean scintillation bandwidth Δνd(1.41 GHz) = 0.21 ± 0.01 MHz, which is attributed to the Milky Way. Balmer line measurements for the host imply an Hαemission measure (galaxy frame) EMs= 620 pc cm−6× (T/104K)0.9, implying DMHαof order the value inferred from the FRB DM budget,DMh=1121138+89pc cm−3for plasma temperatures greater than the typical value 104K. Combiningτand DMhyields a nominal constraint on the scattering amplification from the host galaxyF˜G=1.50.3+0.8(pc2km)1/3, whereF˜describes turbulent density fluctuations andGrepresents the geometric leverage to scattering that depends on the location of the scattering material. For a two-screen scattering geometry whereτarises from the host galaxy and Δνdfrom the Milky Way, the implied distance between the FRB source and dominant scattering material is ≲100 pc. The host galaxy scattering and DM contributions support a novel technique for estimating FRB redshifts using theτ–DM relation, and are consistent with previous findings that scattering of localized FRBs is largely dominated by plasma within host galaxies and the Milky Way.

    more » « less
  4. Abstract

    A classical parking function of lengthnis a list of positive integers$$(a_1, a_2, \ldots , a_n)$$(a1,a2,,an)whose nondecreasing rearrangement$$b_1 \le b_2 \le \cdots \le b_n$$b1b2bnsatisfies$$b_i \le i$$bii. The convex hull of all parking functions of lengthnis ann-dimensional polytope in$${\mathbb {R}}^n$$Rn, which we refer to as the classical parking function polytope. Its geometric properties have been explored in Amanbayeva and Wang (Enumer Combin Appl 2(2):Paper No. S2R10, 10, 2022) in response to a question posed by Stanley (Amer Math Mon 127(6):563–571, 2020). We generalize this family of polytopes by studying the geometric properties of the convex hull of$${\textbf{x}}$$x-parking functions for$${\textbf{x}}=(a,b,\dots ,b)$$x=(a,b,,b), which we refer to as$${\textbf{x}}$$x-parking function polytopes. We explore connections between these$${\textbf{x}}$$x-parking function polytopes, the Pitman–Stanley polytope, and the partial permutahedra of Heuer and Striker (SIAM J Discrete Math 36(4):2863–2888, 2022). In particular, we establish a closed-form expression for the volume of$${\textbf{x}}$$x-parking function polytopes. This allows us to answer a conjecture of Behrend et al. (2022) and also obtain a new closed-form expression for the volume of the convex hull of classical parking functions as a corollary.

    more » « less
  5. For each odd integern≥<#comment/>3n \geq 3, we construct a rank-3 graphΛ<#comment/>n\Lambda _nwith involutionγ<#comment/>n\gamma _nwhose realC∗<#comment/>C^*-algebraCR∗<#comment/>(Λ<#comment/>n,γ<#comment/>n)C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda _n, \gamma _n)is stably isomorphic to the exotic Cuntz algebraEn\mathcal E_n. This construction is optimal, as we prove that a rank-2 graph with involution(Λ<#comment/>,γ<#comment/>)(\Lambda ,\gamma )can never satisfyCR∗<#comment/>(Λ<#comment/>,γ<#comment/>)∼<#comment/>MEEnC^*_{\scriptscriptstyle \mathbb {R}}(\Lambda , \gamma )\sim _{ME} \mathcal E_n, and Boersema reached the same conclusion for rank-1 graphs (directed graphs) in [Münster J. Math.10(2017), pp. 485–521, Corollary 4.3]. Our construction relies on a rank-1 graph with involution(Λ<#comment/>,γ<#comment/>)(\Lambda , \gamma )whose realC∗<#comment/>C^*-algebraCR∗<#comment/>(Λ<#comment/>,γ<#comment/>)C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda , \gamma )is stably isomorphic to the suspensionSRS \mathbb {R}. In the Appendix, we show that theii-fold suspensionSiRS^i \mathbb {R}is stably isomorphic to a graph algebra iff−<#comment/>2≤<#comment/>i≤<#comment/>1-2 \leq i \leq 1.

    more » « less