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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]

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]

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]

Given a nonnegative function ..., we define the Takagi type function ... where dZ(x) := dist(x, Z) := inf{|x - k| : k ∈ Z}. The main result of the paper states that if φ(0) = 0 and the mapping x → φ(x)/x is concave, then the Takagi type function Sφ is approximately Jensen convex in the following sense: ... Applications to the theory of approximately convex functions are also given. [ABSTRACT FROM AUTHOR]

A characterization of convex functions in RN states that an upper semicontinuous function u is convex if and only if u(Ax) is subharmonic (with respect to the usual Laplace operator) for every symmetric positive definite matrix A. The aim of this paper is to prove that an analogue of this result holds for free Carnot groups G when considering convexity in the viscosity sense. In the subelliptic context of Carnot groups, the linear maps x … Ax of the Euclidean case must be replaced by suitable group isomorphisms x … TA(x), whose differential preserves the first layer of the stratification of Lie(G). [ABSTRACT FROM AUTHOR]

In this paper we prove that for every unbounded convex closed set C in c0 there exists a nonexpansive mapping T : C → C which is fixed point free. This result solves in a negative sense a question that has remained open for some time in Metric Fixed Point Theory. [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]

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]

Consider the intrinsic fundamental group à la Grothendieck of a linear category, introduced in our earlier papers using connected gradings. In this article we prove that any full convex subcategory is incompressible, in the sense that the group map between the corresponding fundamental groups is injective. We start by proving the functoriality of the intrinsic fundamental group with respect to full subcategories, based on the study of the restriction of connected gradings. [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 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 study the smallest singular value of a square random matrix with i.i.d. columns drawn from an isotropic log-concave distribution. An important example is obtained by sampling vectors uniformly distributed in an isotropic convex body. We deduce that the condition number of such matrices is of the order of the size of the matrix and give an estimate on its tail behaviour. [ABSTRACT FROM AUTHOR]

In this paper we prove that for every unbounded convex closed set C in c0 there exists a nonexpansive mapping T : C → C which is fixed point free. This result solves in a negative sense a question that has remained open for some time in Metric Fixed Point Theory. [ABSTRACT FROM AUTHOR]

In the works of Darboux and Walsh, it was remarked that a one-to-one self-mapping of $mathbb {R}^{3}$ which sends convex sets to convex ones is affine. It can be remarked also that a $mathcal {C}^{2}$-diffeomorphism $F:Uto U^{'}$ between two domains in $mathbb {C}^{n}$, $nge 2$, which sends pseudoconvex hypersurfaces to pseudoconvex ones is either holomorphic or antiholomorphic. par In this paper we are interested in the self-mappings of $mathbb {C}^{n}$ which send convex hypersurfaces to pseudoconvex ones. Their characterization is the following: {em A } $mathcal {C}^{2}$-{em diffeomorphism} $F:U'to U$ {em (where } $U', Usubset mathbb {C}^{n}$ {em are domains) sends convex hypersurfaces to pseudoconvex ones if and only if the inverse map} $Phi := F^{-1}$ {em is weakly pluriharmonic, } i.e., {em if it satisfies some nice second order PDE very close to} $partial bar {partial }Phi = 0$. In fact all pluriharmonic $Phi $'s do satisfy this equation, but there are also other solutions. [ABSTRACT FROM AUTHOR]

A real valued function $f$ defined on a real interval $I$ is called $(\varepsilon,\delta)$-convex if it satisfies \[ f(tx+(1-t)y)\le tf(x)+(1-t)f(y) + \varepsilon t(1-t)|x-y| + \delta \quad \text{for} x,y\in I, t\in[0,1]. \] The main results of the paper offer various characterizations for $(\varepsilon,\delta)$-convexity. One of the main results states that $f$ is $(\varepsilon,\delta)$-convex for some positive $\varepsilon$ and $\delta$ if and only if $f$ can be decomposed into the sum of a convex function, a function with bounded supremum norm, and a function with bounded Lipschitz-modulus. In the special case $\varepsilon=0$, the results reduce to that of Hyers, Ulam, and Green obtained in 1952 concerning the so-called $\delta$-convexity. [ABSTRACT FROM AUTHOR]

Let $\Phi (t)$ and $\Psi (t)$ be nonnegative convex functions, and let $\varphi $ and $\psi $ be the right continuous derivatives of $\Phi $ and $\Psi ,$ respectively. In this paper, we prove the equivalence of the following three conditions: (i) $\|f^{*}\| _\Phi \leq c\|f\|_\Psi ,$ (ii) $L^\Psi \subseteq $ $H^\Phi $ and (iii) $\int_{s_0}^t\frac{\varphi (s)}sds\leq c\psi (ct), \forall t>s_0,$ where $L^\Psi $ and $H^\Phi $ are the Orlicz martingale spaces. As a corollary, we get a sufficient and necessary condition under which the extension of Doob's inequality holds. We also discuss the converse inequalities. [ABSTRACT FROM AUTHOR]

This paper presents a necessary and sufficient condition on the convex function $f$ in order that continuous solutions to \[\hbox {minimize} \int _{\Omega } f(\|\nabla u(x)\|) dx \hbox { on } u^{0} + W^{1,1}_{0}(\Omega )\] satisfy a Strong Maximum Principle on any open connected $\Omega $. [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]

Let $X$ be a real, reflexive, locally uniformly convex Banach space with $X^{*}$ locally uniformly convex. ~Let $T:X\supset D(T)\to 2^{X^{*}}$ be a maximal monotone operator and $G\subset X$ open and bounded. ~Assume that $M\subset X^{*}$ is pathwise connected and such that $T(D(T)\cap G)\cap M \not = \emptyset $ and $ \overline{T(D(T)\cap \partial G)}\cap M = \emptyset .$ ~Then $M\subset T(D(T)\cap G).$ ~If, moreover, $T$ is of type ($S$) on $\partial G,$ then $ \overline{T(D(T)\cap \partial G)}$ may be replaced above by $T(D(T)\cap \partial G).$ The significance of this result lies in the fact that it holds for multi-valued mappings $T$ which do not have to satisfy $\text{int}D(T) \not = \emptyset .$ ~It has also been used in this paper in order to establish a general ``invariance of domain'' result for maximal monotone operators, and may be applied to a greater variety of problems involving partial differential equations. ~No degree theory has been used. ~In addition to the above, necessary and sufficient conditions are given for the existence of a zero (in an open and bounded set $G$) of a completely continuous perturbation $T+C$ of a maximal monotone operator $T$ such that $T+C$ is locally monotone on $G.$ [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]

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]

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]

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]

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]

We extend Dudley's characterization of convex functions to the framework of h-convex distributions on stratified groups. Precisely, we prove that every distribution with nonnegative horizontal Hessian is defined by an h-convex function. [ABSTRACT FROM AUTHOR]

Given x0, a point of a convex subset C of a Euclidean space, the two following statements are proven to be equivalent: (i) every convex function f : C → ℝ is upper semi-continuous at x0, and (ii) C is polyhedral at x0. In the particular setting of closed convex functions and Fs domains, we prove that every closed convex function f : C → ℝ is continuous at x0 if and only if C is polyhedral at x0. This provides a converse to the celebrated Gale-Klee-Rockafellar theorem. [ABSTRACT FROM AUTHOR]

For a sequence (fn)n?N of nonnegative random variables, we provide simple necessary and sufficient conditions for convergence in probability of each sequence (hn)n?N with hn ? conv({fn, fn+1, . . . }) for all n ? N to the same limit. These conditions correspond to an essentially measure-free version of the notion of uniform integrability. [ABSTRACT FROM AUTHOR]

Brunn-Minkowski type inequalities for radial Blaschke-Minkowski homomorphisms of star bodies and polar Blaschke-Minkowski homomorphisms of convex bodies are established. [ABSTRACT FROM AUTHOR]

We define a computable topological invariant μ(γ) for generic closed planar regular curves γ, which gives an effective lower bound for the number of inflection points on a given generic closed planar curve. Using it, we classify the topological types of locally convex curves (i.e. closed planar regular curves without inflections) whose numbers of crossings are less than or equal to five. Moreover, we discuss the relationship between the number of double tangents and the invariant μ(γ) of a given γ. [ABSTRACT FROM AUTHOR]

Let Φ : [0,∞) → R be a continuous convex function with Φ(0) = 0. We prove that Φ(∥f∥1/ωN∥f∥ ∞)⩽ = 1 ∫RN/ωN∥f∥∞ RN ∣f(x)∣Φ(∣x∣N)dx for every f ∈ L1(RN) ∩ L∞(RN), f ≠ 0, where N is the measure of the unit ball of RN. This can be used to obtain lower or upper bounds for weighted integrals ∫RN ∣f(x)∣η(∣x∣)dx in terms of the L1 and L∞ norms of f, which are often much sharper than crude estimates that may be obtained, if at all, by a visual inspection of the integrand. The basic inequality is essentially independent of Jensen's inequality, but it is closely related to Steffensen's inequality. [ABSTRACT FROM AUTHOR]

In 1981 Walter Rudin exhibited a totally real embedding of the Klein bottle into C2. We show that the polynomially convex hull of Rudin's Klein bottle contains an open subset of C2. We also describe another totally real Klein bottle in C2 whose hull has topological dimension equal to three. [ABSTRACT FROM AUTHOR]