相关论文: John-type theorems for generalized arithmetic prog…
We introduce a family of symmetric convex bodies called generalized ellipsoids of degree $d$ (GE-$d$s), with ellipsoids corresponding to the case of $d=0$. Generalized ellipsoids (GEs) retain many geometric, algebraic, and algorithmic…
We give a representation theoretical proof of Branson's classification of minimal elliptic sums of generalized gradients. The original proof uses tools of harmonic analysis, which as powerful as they are, seem to be specific for the…
Some sorts of generalized morphisms are defined from very basic mathematical objects such as sets, functions, and partial functions. A wide range of mathematical notions such as continuous functions between topological spaces, ring…
Taking as model the attractor of an iterated function system consisting of phi-contractions on a complete and bounded metric space, we introduce the set-theoretic concept of family of functions having attractor. We prove that, given such a…
In the paper, we introduce the generalized convex function on fractal sets of real line numbers and study the properties of the generalized convex function. Based on these properties, we establish the generalized Jensen inequality and…
It is shown that Alesker's solution of McMullen's conjecture implies the following stronger version of the conjecture: Every continuous, translation invariant, $k$-homogeneous valuation on convex bodies in $\mathbb{R}^n$ can be approximated…
If $a$ and $b$ are integers with $b>a>1$, we completely characterize ``long'' arithmetic progressions in the sumsets of the geometric progressions $1, a, a^2, a^3, \ldots$ and $1, b, b^2, b^3, \ldots$. Our proofs utilize recent applications…
We consider two well-known problems: upper bounding the volume of lower dimensional ellipsoids contained in convex bodies given their John ellipsoid, and lower bounding the volume of ellipsoids containing projections of convex bodies given…
In this paper, we address the well-posedness theory of F. John's problem for freely floating objects in a two-dimensional framework. This problem is a linear description of the interactions between an incompressible, irrotational…
Alesker's theory of generalized valuations unifies smooth measures and constructible functions on real analytic manifolds, extending classical operations on functions and measures. Alesker showed that these operations agree with the…
Let $AP_k=\{a,a+d,\ldots,a+(k-1)d\}$ be an arithmetic progression. For $\epsilon>0$ we call a set $AP_k(\epsilon)=\{x_0,\ldots,x_{k-1}\}$ an $\epsilon$-approximate arithmetic progression if for some $a$ and $d$, $|x_i-(a+id)|<\epsilon d$…
The polynomial Fre\u{\i}man--Ruzsa conjecture is a fundamental open question in additive combinatorics. However, over the integers (or more generally $\mathbb{R}^d$ or $\mathbb{Z}^d$) the optimal formulation has not been fully pinned down.…
We prove a conjecture of B. Gr\"unbaum stating that the set of affine invariant points of a convex body equals to the set of points invariant under all affine linear symmetries of the convex body. As a consequence we give a short proof on…
The Classical Jacobian Conjecture claims that any unramified endomorphism of a complex affine space is an automorphism. In order to embed this conjecture in a geometric environment, where one could enjoy the beauty and the richness of tools…
In this article, we consider limit theorems for some weighted type random sums (or discrete rough integrals). We introduce a general transfer principle from limit theorems for unweighted sums to limit theorems for weighted sums via rough…
The tensorial curvature measures are tensor-valued generalizations of the curvature measures of convex bodies. On convex polytopes, there exist further generalizations some of which also have continuous extensions to arbitrary convex…
This article derives closed-form parametric formulas for the Minkowski sums of convex bodies in d-dimensional Euclidean space with boundaries that are smooth and have all positive sectional curvatures at every point. Under these conditions,…
The paper is devoted to a comprehensive second-order study of a remarkable class of convex extended-real-valued functions that is highly important in many aspects of nonlinear and variational analysis, specifically those related to…
This paper explores formalizing Geometric (or Clifford) algebras into the Lean 3 theorem prover, building upon the substantial body of work that is the Lean mathematics library, mathlib. As we use Lean source code to demonstrate many of our…
In the recently introduced gauge theory of translations, dubbed Coincident General Relativity, gravity is described with neither torsion nor curvature in the spacetime affine geometry. The action of the theory enjoys an enhanced symmetry…