Mathematika最新文献

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 their Loewner ellipsoid. For the first problem, we use the John asymmetry to unify a tight upper bound for the general case by Ball with a stronger inequality for symmetric convex bodies. We obtain an inequality that is tight for most asymmetry values in large dimensions and an even stronger inequality in the planar case that is always best possible. In contrast, we show for the second problem an inequality that is tight for bodies of any asymmetry, including cross-polytopes, parallelotopes, and (in almost all cases) simplices. Finally, we derive some consequences for the width-circumradius- and diameter-inradius-ratios when optimized over affine transformations and show connections to the Banach–Mazur distance.

We give a soft proof of a uniform upper bound for the local factors in the triple product formula, sufficient for deducing effective and general forms of quantum unique ergodicity (QUE) from subconvexity.

Let $$ be a probability measure on $$. We give conditions on the Fourier transform of its density for functionals of the form $$ to be Schur monotone. As applications, we put certain known and new results under the same umbrella, given by a condition on the Fourier transform of the density. These results include certain moment comparisons for independent and identically distributed random vectors, when the norm is given by intersection bodies, and vector-valued analogues of Khinchin's inequality with respect to appropriate norms. We also extend the discussion to higher dimensions.

Let $$ be an elliptic curve defined over $$, and let $$ be an imaginary quadratic field. Consider an odd prime $$ at which $$ has good supersingular reduction with $$ and which is inert in $$. Under the assumption that the signed Selmer groups are cotorsion modules over the corresponding Iwasawa algebra, we prove that the Mordell–Weil ranks of $$ are bounded over any subextensions of the anticyclotomic $$-extension of $$. Additionally, we provide an asymptotic formula for the growth of the $$-parts of the Tate–Shafarevich groups of $$ over these extensions.

We prove the direct and the converse inequalities for type IV superorthogonality in the vector-valued setting. The converse one is also new in the scalar setting.

We study the $$ moment of central values of the family of Dirichlet $$-functions to a fixed prime modulus and establish sharp upper bounds for all real $$.

In Euclidean space, the asymptotic shape of large cells in various types of Poisson-driven random tessellations has been the subject of a famous conjecture due to David Kendall. Since shape is a geometric concept and large cells are identified by means of geometric size functionals, the resolution of the conjecture is inevitably connected with geometric inequalities of isoperimetric type and their improvements in the form of geometric stability results, relating geometric size functionals and hitting functionals. The latter are deterministic characteristics of the underlying random tessellation. The current work explores specific and typical cells of random tessellations in spherical space. A key ingredient of our approach is new geometric inequalities and quantitative strengthenings in terms of stability results for general and also for some specific size and hitting functionals of spherically convex bodies. As a consequence, we obtain probabilistic deviation inequalities and asymptotic distributions of quite general size functionals. In contrast to the Euclidean setting, where naturally the asymptotic regime concerns large size, in the spherical framework, the asymptotic analysis is primarily concerned with high intensities.

In this article, we study discrete maximal function associated with the Birch–Magyar averages over sparse sequences. We establish sparse domination principle for such operators. As a consequence, we obtain $$-estimates for such discrete maximal functions over sparse sequences for all $$. The proof of sparse bounds is based on scale-free $$-improving estimates for the single scale Birch–Magyar averages.

Three new combinations of convex bodies are introduced and studied: the $$ fiber, $$ chord, and graph combinations. These combinations are defined in terms of the fibers and graphs of pairs of convex bodies, and each operation generalizes the classical Steiner symmetral, albeit in different ways. For the $$ fiber and $$ chord combinations, we derive Brunn–Minkowski-type inequalities and the corresponding Minkowski's first inequalities. We also prove that the general affine surface areas are concave (respectively, convex) with respect to the graph sum, thereby generalizing fundamental results of Ye (Indiana Univ. Math. J. 14 (2014), 1–19) on the monotonicity of the general affine surface areas under Steiner symmetrization. As an application, we deduce a corresponding Minkowski's first inequality for the $$ affine surface area of a graph combination of convex bodies.

Motivated by general probability theory, we say that the set $$ in $$ is antipodal of rank $$, if for any $$ elements $$, there is an affine map from $$ to the $$-dimensional simplex $$ that maps $$ bijectively onto the $$ vertices of $$. For $$, it coincides with the well-studied notion of (pairwise) antipodality introduced by Klee. We consider the following natural generalization of Klee's problem on antipodal sets: What is the maximum size of an antipodal set of rank $$ in $$? We present a geometric characterization of antipodal sets of rank $$ and adapting the argument of Danzer and Grünbaum originally developed for the $$ case, we prove an upper bound which is exponential in the dimension. We show that this problem can be connected to a classical question in computer science on finding perfect hashes, and it provides a lower bound on the maximum size, which is also exponential in the dimension. By connecting rank-$$ antipodality to $$-neighborly polytopes, we obtain another upper bound when $$.