Related papers: On the concavity of the arithmetic volumes
We extend some part of the unpublished paper written by Mednykh and Rasskazov. Using the approach indicated in this paper we derive the Riley-Mednykh polynomial for some family of the $2$-bridge knot orbifolds. As a result we obtain…
On the class of log-concave functions on $\R^n$, endowed with a suitable algebraic structure, we study the first variation of the total mass functional, which corresponds to the volume of convex bodies when restricted to the subclass of…
Mixed volumes $V(K_1,\dots, K_d)$ of convex bodies $K_1,\dots ,K_d$ in Euclidean space $\mathbb{R}^d$ are of central importance in the Brunn-Minkowski theory. Representations for mixed volumes are available in special cases, for example as…
In this paper we give simple proofs for the bounds (some of them sharp) of the difference of the moduli of the second and the first logarithmic coefficient for the general class of univalent functions and for the class of convex univalent…
We study the algebraic rank of a divisor on a graph, an invariant defined using divisors on algebraic curves dual to the graph. We prove it satisfies the Riemann-Roch formula, a specialization property, and the Clifford inequality. We prove…
In "Weighted Brunn-Minkowski Theory I", the prequel to this work, we discussed how recent developments on concavity of measures have laid the foundations of a nascent weighted Brunn-Minkowski theory. In particular, we defined the mixed…
We associate convex regions in R^n to m-primary graded sequences of subspaces, in particular m-primary graded sequences of ideals, in a large class of local algebras (including analytically irreducible local domains). These convex regions…
We prove that for any semi-norm $\|\cdot\|$ on $\mathbb{R}^n,$ and any symmetric convex body $K$ in $\mathbb{R}^n,$ \begin{equation}\label{ineq-abs2} \int_{\partial K} \frac{\|n_x\|^2}{\langle x,n_x\rangle}\leq…
A determination of the fixed components, base points and irregularity is made for arbitrary numerically effective divisors on any smooth projective rational surface having an effective anticanonical divisor. All of the results are proven…
We demonstrate the generic invariance of the Fano type property in cases where the volumes of anti-canonical divisors of Fano type fibers are a constant over a Zariski-dense subset, or the Fano type fibers are dimension $2$. Additionally,…
Some mathematical inequalities among various weighted means are studied. Inequalities on weighted logarithmic mean are given. Besides, the gap in Jensen's inequality is studied as a convex function approach. Consequently, some non-trivial…
It is shown that each continuous even Minkowski valuation on convex bodies of degree $1 \leq i \leq n - 1$ intertwining rigid motions is obtained from convolution of the $i$th projection function with a unique spherical Crofton…
This paper proves the volume of the arithmetic Okounkov body, constructed from a hermitian line bundle on an arithmetic variety by the author in a previous paper, is equal to the the volume of the hermitian line bundle up to a simple…
We study a class of semialgebraic convex bodies called discotopes. These are instances of zonoids, objects of interest in real algebraic geometry and random geometry. We focus on the face structure and on the boundary hypersurface of…
We prove explicit upper bounds for weighted sums over prime numbers in arithmetic progressions with slowly varying weight functions. The results generalize the well-known Brun-Titchmarsh inequality.
The Minkowski inequality is a classical inequality in differential geometry, giving a bound from below, on the total mean curvature of a convex surface in Euclidean space, in terms of its area. Recently there has been interest in proving…
In this article, we determine the Rogosinski radii for certain subclasses of close-to-convex functions defined on open unit disc $\mathbb{D}= \{z \in \mathbb{C}: |z| < 1\}$. Furthermore, we establish improved versions of the classical Bohr…
The purpose of this paper is to initiate Arakelov theory in a noncommutative setting. More precisely, we are concerned with noncommutative arithmetic surfaces. We introduce a version of arithmetic intersection theory on noncommutative…
In this paper we study normal surfaces whose anticanonical divisors are strictly nef, i.e. (-K)C>0 for every curve C.
Let $X$ be a smooth projective toric variety over $Spec(Z)$. Let $\bar{\mathcal{O}(D)}$ be an equivariant hermitian line bundle on $X$, equipped with a positive metric and invariant under the action of the compact torus of $X(C)$, we…