A sharp lower bound for the log canonical threshold

In this note, we prove a sharp lower bound for the log canonical threshold of a plurisubharmonic function ${\varphi}$ with an isolated singularity at 0 in an open subset of ${\mathbb{C}^n}$ . This threshold is defined as the supremum of constants c > 0 such that ${e^{-2c\varphi}}$ is integrable on a neighborhood of 0. We relate ${c(\varphi)}$ to the intermediate multiplicity numbers ${e_j(\varphi)}$ , defined as the Lelong numbers of ${(dd^c\varphi)^j}$ at 0 (so that in particular ${e_0(\varphi)=1}$ ). Our main result is that ${c(\varphi)\geqslant\sum_{j=0}^{n-1} e_j(\varphi)/e_{j+1}(\varphi)}$ . This inequality is shown to be sharp; it simultaneously improves the classical result ${c(\varphi)\geqslant 1/e_1(\varphi)}$ due to Skoda, as well as the lower estimate ${c(\varphi)\geqslant n/e_n(\varphi)^{1/n}}$ which has received crucial applications to birational geometry in recent years. The proof consists in a reduction to the toric case, i.e. singularities arising from monomial ideals.     

A sharp lower bound for the first eigenvalue on Finsler manifolds

In this paper, we give a sharp lower bound for the first (nonzero) Neumann eigenvalue of Finsler-Laplacian in Finsler manifolds in terms of diameter, dimension, weighted Ricci curvature.     

A lower bound for the volume of a four-dimensional closed manifold

A closed orientable four-dimensional manifoldM ⁴, whose sectional curvatures satisfy –1 K 1, is considered. If the product of the curvatures of orthogonal area elements is nonnegative and if at least at one point all the curvatures are different from zero, then the estimate vol(M ⁴) > (4/9)² is obtained for the volume ofM ⁴. A theorem on the local structure of a manifold with small volume whose curvatures at every point are of the same sign is established.Translated from Ukrainskii Geometricheskii Sbornik, No. 34, pp. 3–8, 1991.     

A lower bound of cohomologic dimension for an elliptic space

Our goal in this paper is to prove that, under appropriate hypotheses, the sum of the Betti numbers of a 1-connected elliptic space is greater or equal to the dimension of its Q vector space of homotopy. The paper concludes with some examples for which the inequality is strict.     

A sharp lower bound for the canonical volume of 3-folds of general type

Let V be a smooth projective 3-fold of general type. Denote by K ³, a rational number, the self-intersection of the canonical sheaf of any minimal model of V. One defines K ³ as the canonical volume of V. Assume p _g (V) ≥ 2. We show that , which is a sharp lower bound. Then we classify those V with small volume. We also give some new examples with p _g  = 2 which have maximal canonical stability index. Finally, we give an application to 4-folds of general type. The author is supported by the National Natural ScienceFoundation of China.     

A lower bound on the number of sharp shadow-boundaries of convex polytopes

We give the lower bound on the number of sharp shadow-boundaries of convexd-polytopes (or unbounded convex polytopal sets) withn facets. The polytopes (sets) attaining these bounds are characterized. Additionally, our results will be transferred to the dual theory.The research work of the first author was (partially) supported by Hungarian National Foundation for Scientific Research, grant no. 1812.     

A sharp upper bound for the first eigenvalue of the Laplacian of compact hypersurfaces in rank-1 symmetric spaces

Let M be a closed hypersurface in a simply connected rank-1 symmetric space . In this paper, we give an upper bound for the first eigenvalue of the Laplacian of M in terms of the Ricci curvature of and the square of the length of the second fundamental form of the geodesic spheres with center at the center-of-mass of M.     

A lower bound for the worst-case cubature error on spheres of arbitrary dimension

This paper is concerned with numerical integration on the unit sphere S^r of dimension r≥2 in the Euclidean space ℝ^r+1. We consider the worst-case cubature error, denoted by E(Q_m;H^s(S^r)), of an arbitrary m-point cubature rule Q_m for functions in the unit ball of the Sobolev space H^s(S^r), where s>, and show that The positive constant c_s,r in the estimate depends only on the sphere dimension r≥2 and the index s of the Sobolev space H^s(S^r). This result was previously only known for r=2, in which case the estimate is order optimal. The method of proof is constructive: we construct for each Q_m a `bad' function f_m, that is, a function which vanishes in all nodes of the cubature rule and for which Our proof uses a packing of the sphere S^r with spherical caps, as well as an interpolation result between Sobolev spaces of different indices.     

A sharp bound on the Hausdorff dimension of the singular set of a uniform measure

The study of the geometry of n-uniform measures in \(\mathbb {R}^{d}\) has been an important question in many fields of analysis since Preiss’ seminal proof of the rectifiability of measures with positive and finite density. The classification of uniform measures remains an open question to this day. In fact there is only one known example of a non-trivial uniform measure, namely 3-Hausdorff measure restricted to the Kowalski–Preiss cone. Using this cone one can construct an n-uniform measure whose singular set has Hausdorff dimension \(n-3\). In this paper, we prove that this is the largest the singular set can be. Namely, the Hausdorff dimension of the singular set of any n-uniform measure is at most \(n-3\).     

A sharp lower bound on Steiner Wiener index for trees with given diameter

Let $G$ be a connected graph with vertex set $V\left(G\right)$ and edge set $E\left(G\right)$. For a subset $S$ of $V\left(G\right)$, the Steiner distance$d\left(S\right)$ of $S$ is the minimum size of a connected subgraph whose vertex set contains $S$. For an integer $k$ with $2\le k\le n?1$, the Steiner$k$-Wiener index${\text{SW}}_k\left(G\right)$ is ${\sum }_{S?V\left(G\right),\left|S\right|=k}d\left(S\right)$. In this paper, we introduce some transformations for trees that do not increase their Steiner $k$-Wiener index for $2\le k\le n?1$. Using these transformations, we get a sharp lower bound on Steiner $k$-Wiener index for trees with given diameter, and obtain the corresponding extremal graph as well.     

常维码的一个构造性下界

推广了Etzion和Vardy关于常维码的结论（Etzion T,Vardy A.Error-correcting codes in projective space.IEEE Transactions on Information Theory,2011,57（2）：1165-1173）,给出了一般情况下常维码的一个构造性下界.     

A sharp bound for the Stein-Wainger oscillatory integral

Let denote the space of all real polynomials of degree at most . It is an old result of Stein and Wainger that

for some constant depending only on . On the other hand, Carbery, Wainger and Wright claim that the true order of magnitude of the above principal value integral is . We prove that

     

A sharp isoperimetric bound for convex bodies

We consider the problem of lower bounding the Minkowski content of subsets of a convex body with a log-concave probability measure, conditioned on the set size. A bound is given in terms of diameter and set size, which is sharp for all set sizes, dimensions, and norms. In the case of uniform density a stronger theorem is shown which is also sharp. Supported in part by VIGRE grants at Yale University and the Georgia Institute of Technology.