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Let \mathcal {B} be the collection of rectangular parallelepipeds in \mathbb {R}^3 whose sides are parallel to the coordinate axes and such that \mathcal {B} consists of parallelepipeds with side lengths of the form s, 2^j s, t, where s, t > 0 and j lies in a nonempty subset S of the integers. In this paper, we prove the following: If S is a finite set, then the associated geometric maximal operator M_\mathcal {B} satisfies the weak type estimate \begin{equation*} \left |\left \{x \in \mathbb {R}^3 : M_{\mathcal {B}}f(x) > \alpha \right \}\right | \leq C \int _{\mathbb {R}^3} \frac {|f|}{\alpha }\left (1 + \log ^+ \frac {|f|}{\alpha }\right)\; \end{equation*} but does not satisfy an estimate of the form \begin{equation*} \left |\left \{x \in \mathbb {R}^3 : M_{\mathcal {B}}f(x) > \alpha \right \}\right | \leq C \int _{\mathbb {R}^3} \phi \left (\frac {|f|}{\alpha }\right) \end{equation*} for any convex increasing function \phi : \mathbb [0, \infty) \rightarrow [0, \infty) satisfying the condition \begin{equation*} \lim _{x \rightarrow \infty }\frac {\phi (x)}{x (\log (1 + x))} = 0. \end{equation*} On the other hand, if S is an infinite set, then the associated geometric maximal operator M_\mathcal {B} satisfies the weak type estimate \begin{equation*} \left |\left \{x \in \mathbb {R}^3 : M_{\mathcal {B}}f(x) > \alpha \right \}\right | \leq C \int _{\mathbb {R}^3} \frac {|f|}{\alpha } \left (1 + \log ^+ \frac {|f|}{\alpha }\right)^{2} \end{equation*} but does not satisfy an estimate of the form \begin{equation*} \left |\left \{x \in \mathbb {R}^3 : M_{\mathcal {B}}f(x) > \alpha \right \}\right | \leq C \int _{\mathbb {R}^3} \phi \left (\frac {|f|}{\alpha }\right) \end{equation*} for any convex increasing function \phi : \mathbb [0, \infty) \rightarrow [0, \infty) satisfying the condition \begin{equation*} \lim _{x \rightarrow \infty }\frac {\phi (x)}{x (\log (1 + x))^2} = 0. \end{equation*} [ABSTRACT FROM AUTHOR]

In this paper we establish a new identity, and then based on this identity we derive the Maclaurin's inequality for multiplicatively convex functions. [ABSTRACT FROM AUTHOR]

Recently, Simons provided a lemma for a support function of a closed convex set in a general Hilbert space and used it to prove the geometry conjecture on cycles of projections. In this paper, we extend Simons's lemma to closed convex functions, show its connections to Attouch–Théra duality, and use it to characterize (classical and phantom) cycles and gap vectors of proximal mappings. [ABSTRACT FROM AUTHOR]

In this paper, we provide some algebraic structures of convex subrings of *C, a nonstandard extension of the field of complex numbers C. In particular, a detailed description of the prime spectrum of any convex subring of *C is given. To achieve our goal, first we investigate prime ideals and we characterize two consecutive elements in the spectrum of a divided domain. We also show that the prime spectrum of the ring of bounded hypercomplex numbers has two peculiar properties: there are no three consecutive elements in the spectrum; moreover, nonzero elements are a disjoint union of three subsets where one of them is strongly dense and the other two are dense in the spectrum. [ABSTRACT FROM AUTHOR]

In this paper, we establish some conformable fractional Hermite-Hadamard type integral inequalities via harmonic s-convexity, and the estimates of the products of two harmonic s-convex functions are also considered. [ABSTRACT FROM AUTHOR]

In this paper we consider numerical approximation to the generalised solutions to the Monge-Ampère type equations with general source terms. We first give some important propositions for the border of generalised solutions. Then, for both the classical and weak Dirichlet boundary conditions, we present well-posed numerical methods for the generalised solutions with general source terms. Finally, we prove that the numerical solutions converge to the generalised solution. [ABSTRACT FROM AUTHOR]

Let I ⊂ N be an infinite subset, and let {ai}i≤I be a sequence of nonzero real numbers indexed by I such that there exist positive constants m,C1 for which |ai| ≤ C1·im for all I ∈ i. Furthermore, let ci ∈ [-1, 1] be defined by ci = ai C1·im for each i ∈ I, and suppose the ci's are equidistributed in [-1, 1] with respect to a continuous, symmetric probability measure μ. In this paper, we show that if I ⊂ N is not too sparse, then the sequence {ai}i∈I fails to obey Benford's Law with respect to arithmetic density in any sufficiently large base, and in fact in any base when μ([0, t]) is a strictly convex function of t ∈ (0, 1). Nonetheless, we also provide conditions on the density of I ⊂ N under which the sequence {ai}i∈I satisfies Benford's Law with respect to logarithmic density in every base. As an application, we apply our general result to study Benford's Law-type behavior in the leading digits of Frobenius traces of newforms of positive, even weight. Our methods of proof build on the work of Jameson, Thorner, and Ye, who studied the particular case of newforms without complex multiplication. [ABSTRACT FROM AUTHOR]

The objective of this paper is to investigate operator functions by making use of the operator harmonic mean ' !'. For 0 < A ≤ B, we construct a unique pair X, Y such that 0 < X ≤ Y, A = X,!Y,; B = X+Y/2. We next give a condition for operators A, B, C ≥ 0 in order that C ≤ A! B and show that g ≠ 0 is strongly operator convex on J if and only if g(t) > 0 and g (A+B/2 ≤ g(A)!g(B) for A, B with spectra in J. This inequality particularly holds for an operator decreasing function on the right half line. We also show that ƒ(t) defined on (0, b) with 0 < b ≤ ∞ is operator monotone if and only if ƒ(0+) < ∞ ƒ (A!B) ≤ 1/2(f(A) + f(B)). In particular, if ƒ > 0, then f is operator monotone if and only if ƒ (A ! B) ≤ ƒ(A) ! ƒ(B). We lastly prove that if a strongly operator convex function g(t) > 0 on a finite interval (a, b) is operator decreasing, then g has an extension g to (a, ∞) that is positive and operator decreasing. [ABSTRACT FROM AUTHOR]

We establish some \Phi-moment inequalities for noncommutative differentially subordinate martingales. Let \Phi be a p-convex and q-concave Orlicz function with 1

In this paper we prove that any C1,1/2curve in Rn is the solution of the gradient flow equation for some C¹ convex function f if and only if it is strongly self-contracted. [ABSTRACT FROM AUTHOR]

The purpose of this paper is twofold. The first is to investigate the Roper-Suffridge extension operator which maps a biholomorhic function f on D to a biholomorphic mapping F on ..., pj ≥ 1, where z0 = (z2, . . ., zn) and λD is the density of the Poincaré metric on a simply connected domain D ⊂ C. We prove this Roper-Suffridge extension operator preserves ε-starlike mapping: if f is ε-starlike, then so is F. As a consequence, we solve a problem of Graham and Kohr in a new method. By introducing the scaling method, the second part is to construct some new convex mappings of domain Ω2,m = {(z1, z2) ∈ C² : |z1|² + |z2|m < 1} with m ≥ 2, which can be applied to discuss the extremal point of convex mappings on the domain. This scaling idea can be viewed as providing an alternative approach to studying convex mappings on Ω2,m. Moreover, we propose some problems. [ABSTRACT FROM AUTHOR]

In this paper, it is shown that a Wulff shape is strictly convex if and only if its convex integrand is of class C¹. Moreover, applications of this result are given. [ABSTRACT FROM AUTHOR]

K. Ball proved a strengthening of the Blaschke-Santaló inequality for unconditional convex bodies in ℝn. In this paper, we characterize the uniqueness of this inequality and establish its functional version. [ABSTRACT FROM AUTHOR]

This paper consists of two parts. One is that for a kind of selfshrinker in a manifold with warped product metric, we prove that under some conditions on ambient space, the mean convex self-shrinker must have parallel second fundamental form. The other one is a generalization of Brendle's Minkowski inequality for weighted mean curvature. [ABSTRACT FROM AUTHOR]

We prove several noncommutative maximal inequalities associated with convex functions, including a Doob type inequality for a convex function of maximal operators on noncommutative martingales, and noncommutative Dunford-Schwartz and Stein maximal ergodic inequalities for a convex function of positive and symmetric positive contractions. The key ingredient in our proofs is a Marcinkiewicz type interpolation theorem for a convex function of maximal operators in the noncommutative setting, which we establish in this paper. These generalize the results of Junge and Xu in the Lp case to the case of convex functions. [ABSTRACT FROM AUTHOR]

In this paper, we study entire translating solutions u(x) to a mean curvature flow equation in Minkowski space. We show that if Σ = {(x, u(x))∣x ϵ Rn} is a strictly spacelike hypersurface, then Σ reduces to a strictly convex rank k soliton in Rk,1 (after splitting off trivial factors) whose "blowdown" converges to a multiple λ ϵ (0, 1) of a positively homogeneous degree one convex function in Rk. We also show that there is nonuniqueness as the rotationally symmetric solution may be perturbed to a solution by an arbitrary smooth order one perturbation. [ABSTRACT FROM AUTHOR]

The inequalities involving the volume of convex bodies play an important role in convex geometry. In this paper, the Rogers and Shephard inequality for the Orlicz difference body of a convex body in the two-dimensional case is established. [ABSTRACT FROM AUTHOR]

We investigate the convexity problem for the Parisi functional defined on the space of the so-called functional ordered parameters in the Sherrington-Kirkpatrick model. In a recent work of Panchenko, it was proved that this functional is convex along one-sided directions with a probabilistic method. In this paper, we will study this problem with a PDE approach that simplifies the original proof and presents more general results. [ABSTRACT FROM AUTHOR]

In this paper, we establish boundary Holder gradient estimates for solutions to the linearized Monge-Ampere equations with Lp (n < p ≤ ∞) right-hand side and C¹, γ boundary values under natural assumptions on the domain, boundary data and the Monge-Ampere measure. These estimates extend our previous boundary regularity results for solutions to the linearized Monge-Ampere equations with bounded right-hand side and C1,1 boundary data. [ABSTRACT FROM AUTHOR]

A deep result of J. Lewis (1983) shows that the polylogarithms Liɑ(z):= σk=1∞ zk/kɑ map the open unit disk D centered at the origin one-to-one onto convex domains for all ɑ ≥ 0. In the present paper this result is generalized to the so-called universal convexity and universal starlikeness (with respect to the origin) in the slit-domain &Lgr;:= C\[1, ∞), introduced by S. Ruscheweyh, L. Salinas and T. Sugawa (2009). This settles a conjecture made in that work and proves, in particular, that Liɑ(z) maps an arbitrary open disk or half-plane in &Lgr; one-to-one onto a convex domain for every ɑ ≥ 1. [ABSTRACT FROM AUTHOR]

Given a Chevalley group \mathcal {G} of classical type and a Borel subgroup \mathcal {B} \subseteq \mathcal {G}, we compute the \Sigma-invariants of the S-arithmetic groups \mathcal {B}(\mathbb {Z}[1/N]), where N is a product of large enough primes. To this end, we let \mathcal {B}(\mathbb {Z}[1/N]) act on a Euclidean building X that is given by the product of Bruhat–Tits buildings X_p associated to \mathcal {G}, where p is a prime dividing N. In the course of the proof we introduce necessary and sufficient conditions for convex functions on CAT(0)-spaces to be continuous. We apply these conditions to associate to each simplex at infinity \tau \subset \partial _\infty X its so-called parabolic building X^{\tau } and to study it from a geometric point of view. Moreover, we introduce new techniques in combinatorial Morse theory, which enable us to take advantage of the concept of essential n-connectivity rather than actual n-connectivity. Most of our building theoretic results are proven in the general framework of spherical and Euclidean buildings. For example, we prove that the complex opposite each chamber in a spherical building \Delta contains an apartment, provided \Delta is thick enough and Aut(\Delta) acts chamber transitively on \Delta. [ABSTRACT FROM AUTHOR]

We prove the sharp quantitative stability for a wide class of weighted isoperimetric inequalities. More precisely, we consider isoperimetric inequalities in convex cones with homogeneous weights. Inspired by the proof of such isoperimetric inequalities through the ABP method (see X. Cabré, X. Ros-Oton, and J. Serra [J. Eur. Math. Soc. (JEMS) 18 (2016), pp. 2971–2998]), we construct a new convex coupling (i.e., a map that is the gradient of a convex function) between a generic set E and the minimizer of the inequality (as in Gromov's proof of the isoperimetric inequality). Even if this map does not come from optimal transport, and even if there is a weight in the inequality, we adapt the methods of [Figalli, Maggi, and Pratelli [Invent. Math. 182 (2010), pp. 167–211] and prove that if E is almost optimal for the inequality then it is quantitatively close to a minimizer up to translations. Then, a delicate analysis is necessary to rule out the possibility of translations. As a step of our proof, we establish a sharp regularity result for restricted convex envelopes of a function that might be of independent interest. [ABSTRACT FROM AUTHOR]

We quantify Peter Scott's theorem that surface groups are locally extended residually finite (LERF) in terms of geometric data. In the process, we will quantify another result by Scott that any closed geodesic in a surface lifts to an embedded loop in a finite cover. [ABSTRACT FROM AUTHOR]

We develop two-scale methods for computing the convex envelope of a continuous function over a convex domain in any dimension. This hinges on a fully nonlinear obstacle formulation (see A. M. Oberman [Proc. Amer. Math. Soc. 135 (2007), pp. 1689–1694]). We prove convergence and error estimates in the max norm. The proof utilizes a discrete comparison principle, a discrete barrier argument to deal with Dirichlet boundary values, and the property of flatness in one direction within the non-contact set. Our error analysis extends to a modified version of the finite difference wide stencil method provided by Oberman [Math. Models Methods Appl. Sci. 18 (2008), pp. 759–780]. [ABSTRACT FROM AUTHOR]

We study positive weak solutions of the quasilinear Lane-Emden equation −Qu = uα in Ω ⊂ Rn, where α ≥ n+2/n−2, for n ≥ 3, is supercritical and the operator Q, known as Finsler-Laplacian or anisotropic Laplacian, is defined by Qu ≔ ∑i=1n ∂/∂xi(F(∇ u)Fξi(∇u)). Here, Fξi = &#8706F/∂μi and F: Rn → [0,+ ∞) is a convex function of C2(Rn\{0}), that satisfies positive homogeneity of first order and other certain assumptions. We prove that the Hausdorff dimension of singular set of u is less than n−2 α+1/α−1. [ABSTRACT FROM AUTHOR]

We prove that convex viscosity solutions to the quadratic Hessian inequality σ2(D2u) ≥ 1 are strictly 2-convex. As a consequence we obtain short proofs of smoothness and interior C2 estimates for convex viscosity solutions to σ2(D2u) = 1, which were proven using different methods in recent works of Guan-Qiu, McGonagle-Song-Yuan, and Shankar-Yuan. [ABSTRACT FROM AUTHOR]

In this note we investigate the regularity of geodesics in the space of convex and plurisubharmonic functions. In the real setting we prove (optimal) local C1,1 regularity. We construct examples which prove that the global C1,1 regularity fails both in the real and complex case in contrast to the Kähler manifold setting. Finally we show a necessary and sufficient conditions for existence of a smooth geodesic between two smooth strictly convex functions. [ABSTRACT FROM AUTHOR]

We prove that upper curvature bounds in the sense of Alexandrov can be improved locally by using appropriate conformal changes. As a new technical tool we derive a generalization to metric spaces and semi-convex functions of the classical differential geometric property that compositions of harmonic maps with convex functions are subharmonic. [ABSTRACT FROM AUTHOR]

We consider coordinate descent methods for minimization of convex quadratic functions, in which exact line searches are performed at each iteration. (This algorithm is identical to Gauss-Seidel on the equivalent symmetric positive definite linear system.) We describe a class of convex quadratic functions for which the random permutations version of cyclic coordinate descent (RPCD) is observed to outperform the standard cyclic coordinate descent (CCD) approach on computational tests, yielding convergence behavior similar to the fully random variant (RCD). A convergence analysis is developed to explain the empirical observations. [ABSTRACT FROM AUTHOR]

Using recent work of Bourgain-Dyatlov (2018) we show that for any convex co-compact hyperbolic surface Strichartz estimates for the Schrodinger equation hold with an arbitrarily small loss of regularity. [ABSTRACT FROM AUTHOR]

Let b ∈ ℝ, let c > 0, let x > 0, and let Gb,c(x) = e-x / xb Pc(x) with Pc(x) = ∞Σ k=0 xk / Γ(c + k) . We prove that Gb,c is completely monotonic on (0,∞) if and only if b ≥ 0 and b+c ≥ 1. Moreover, we present various functional inequalities for Pc. Among others, we show that if c ∈ (0, 1), then, for x, y > 0 we have e < Pc(1/x)x Pc(1/y)y / Pc(1/(x + y))x+y . If c > 1, then the reverse inequality holds for x, y > 0. In both cases, the constant bound e is best possible. [ABSTRACT FROM AUTHOR]

All SL(n) covariant vector valuations on convex polytopes in Rn are completely classified without any continuity assumptions. The moment vector turns out to be the only such valuation if n = 3, while two new functionals show up in dimension two. [ABSTRACT FROM AUTHOR]

Let K and L be two convex bodies in R3, such that their sections by cones {x ∈ R³: x · ξ = t|x|} or non-central planes with a fixed distance from the origin are directly congruent. We prove that if their boundaries are of class C², then K and L coincide. [ABSTRACT FROM AUTHOR]

For an appropriate class of convex functions φ, we study the Fourier extension operator on the surface {(y, |y|² + φ(y)) : y ∈ R²} equipped with projection measure. For the corresponding extension inequality, we compute optimal constants and prove that extremizers do not exist. The main tool is a new comparison principle for convolutions of certain singular measures that holds in all dimensions. Using tools of concentration-compactness flavor, we further investigate the behavior of general extremizing sequences. Our work is directly related to the study of extremizers and optimal constants for Strichartz estimates of certain higher order Schrödinger equations. In particular, we resolve a dichotomy from the recent literature concerning the existence of extremizers for a family of fourth order Schrödinger equations and compute the corresponding operator norms exactly where only lower bounds were previously known. [ABSTRACT FROM AUTHOR]

Given any continuous, lower bounded, and κ-convex function V on a metric measure space (X, d,m) which is infinitesimally Hilbertian and satisfies some synthetic lower bound for the Ricci curvature in the sense of Lott-Sturm-Villani, we prove existence and uniqueness for the (downward) gradient flow for V. Moreover, we prove Lipschitz continuity of the flow w.r.t. the starting point .... [ABSTRACT FROM AUTHOR]

We investigate the closed convex hull of unitary orbits of selfadjoint elements in arbitrary unital C*-algebras. Using a notion of majorization against unbounded traces, a characterization of these closed convex hulls is obtained. Furthermore, for C*-algebras satisfying Blackadar's strict comparison of positive elements by traces or for collections of C*-algebras with a uniform bound on their nuclear dimension, an upper bound for the number of unitary conjugates in a convex combination required to approximate an element in the closed convex hull within a given error is shown to exist. This property, however, fails for certain "badly behaved" simple nuclear C*-algebras. [ABSTRACT FROM AUTHOR]

We propose a semi-discrete scheme for 2D Keller-Segel equations based on a symmetrization reformation, which is equivalent to the convex splitting method and is free of any nonlinear solver. We show that, this new scheme is stable as long as the initial condition does not exceed certain threshold, and it asymptotically preserves the quasi-static limit in the transient regime. Furthermore, we show that the fully discrete scheme is conservative and positivity preserving, which makes it ideal for simulations. The analogical schemes for the radial symmetric cases and the subcritical degenerate cases are also presented and analyzed. With extensive numerical tests, we verify the claimed properties of the methods and demonstrate their superiority in various challenging applications. [ABSTRACT FROM AUTHOR]

Existence results for viscosity solutions to the Dirichlet problem for stationary Hamilton-Jacobi equations, the associated relaxed problem via quasiconvex envelopes, and for minimizers of the corresponding L∞ functionals are obtained for given boundary data in W1,∞. [ABSTRACT FROM AUTHOR]

We prove that in small codimensions, the sections of a convex body in Rn through its centroid are quite symmetric with respect to volume. As a consequence of our estimates we give a positive answer to a problem posed by M. Meyer and S. Reisner regarding convex intersection bodies. [ABSTRACT FROM AUTHOR]

We derive the dual of the relaxed Monge-Kantorovich optimal mass transport problem in L∞ in which one seeks to minimize μess sup(x,y)∊RN×RN c(x, y) over Borel probability measures μ with given marginals P0, P1.Several formulations of the dual problem are obtained using various techniques including quasiconvex duality. We also consider weighted optimal transport in L∞ and we identify the form of the dual in the Lagrangian cost setting for both integral and essential supremum costs along trajectories. Finally, we prove a duality formula that relates a maximization problem which arises naturally in the L∞ calculus of variations with a family of optimal partial transport problems. [ABSTRACT FROM AUTHOR]

We establish subdifferential calculus rules for the sum of convex functions defined on normed spaces. This is achieved by means of a condition relying on the continuity behaviour of the inf-convolution of their corresponding conjugates, with respect to any given topology intermediate between the norm and the weak* topologies on the dual space. Such a condition turns out to also be necessary in Banach spaces. These results extend both the classical formulas by Hiriart-Urruty and Phelps and by Thibault. [ABSTRACT FROM AUTHOR]

For f analytic and close to convex in D = {z : ∣z∣ < 1}, we give sharp estimates for the logarithmic coefficients γn of f defined by log f(z)/z = 2Σ∝n=1 γnzn when n = 1, 2, 3. [ABSTRACT FROM AUTHOR]

We consider the problem to reconstruct a wave speed c ∈ C∞(M) in a domain M ⊂ Rn from acoustic boundary measurements modelled by the hyperbolic Dirichlet-to-Neumann map Λ. We introduce a reconstruction formula for c that is based on the Boundary Control method and incorporates features also from the complex geometric optics solutions approach. Moreover, we show that the reconstruction formula is locally Lipschitz stable for a low frequency component of c-2 under the assumption that the Riemannian manifold (M, c-2dx²) has a strictly convex function with no critical points. That is, we show that for all bounded C² neighborhoods U of c, there is a C¹ neighborhood V of c and constants C,R > 0 such that ... is the Dirichlet-to-Neumann map corresponding to the wave speed |·| is a norm capturing certain regularity properties of the Dirichlet-to-Neumann maps. [ABSTRACT FROM AUTHOR]

We show that the volume entropy of the Hilbert metric on a closed convex projective surface tends to zero as the corresponding Pick differential tends to infinity. The proof is based on the fact, due to Benoist and Hulin, that the Hilbert metric and the Blaschke metric are comparable. [ABSTRACT FROM AUTHOR]

Let R(x) denote the ring of polynomials in g freely noncommuting variables x = (x1, . . . , xg). There is a natural involution . on R(x) determined by x* j = xj and (pq)* = q*p*, and a free polynomial p E R(x) is symmetric if it is invariant under this involution. If X = (X1, . . . , Xg) is a g tuple of symmetric n × n matrices, then the evaluation p(X) is naturally defined and further p.(X) = p(X). In particular, if p is symmetric, then p(X)* = p(X). The main result of this article says if p is symmetric, p(0) = 0 and for each n and each symmetric positive definite n×n matrix A the set {X : A.p(X) > 0} is convex, then p has degree at most two and is itself convex, or .p is a hermitian sum of squares. [ABSTRACT FROM AUTHOR]