If two symmetric convex bodies K and L both have nicely bounded sections, then the intersection of random rotations of K and L is also nicely bounded. For L being a subspace, this main result immediately yields the unexpected existence versus prevalence phenomenon: If K has one nicely bounded section, then most sections of K are nicely bounded. The main result represents a new connection between the local asymptotic convex geometry (study of sections of convex bodies) and the global asymptotic convex geometry (study of convex bodies as a whole). Our method relies on the recent isoperimetry of waists due to M. Gromov [G].

This is the continuation of our earlier article [10]. For any Kähler-Einstein surfaces with positive scalar curvature, if the initial metric has positive bisectional curvature, then we have proved (see [10]) that the Kähler-Ricci flow converges exponentially to a unique Kähler-Einstein metric in the end. This partially answers a long-standing problem in Ricci flow: On a compact Kähler-Einstein manifold, does the Kähler-Ricci flow converge to a Kähler-Einstein metric if the initial metric has positive bisectional curvature? In this article we give a complete affirmative answer to this problem

Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over $\mathbb{C}$. We classify the homomorphisms between $\mathfrak{g}$-modules induced from one-dimensional modules of maximal parabolic subalgebras

Let $m_{12}$, $m_{13}$, …, $m_{n-1,n}$ be the slopes of the ($\genfrac{}{}{0ex}{}{n}{2}$) lines connecting $n$ points in general position in the plane. The ideal $I_n$ of all algebraic relations among the $m_{\mathrm{ij}}$ defines a configuration space called the slope variety of the complete graph. We prove that $I_n$ is reduced and Cohen-Macaulay, give an explicit Gröbner basis for it, and compute its Hilbert series combinatorially. We proceed chiefly by studying the associated Stanley-Reisner simplicial complex, which has an intricate recursive structure. In addition, we are able to answer many questions about the geometry of the slope variety by translating them into purely combinatorial problems concerning the enumeration of trees

We study positive entire solutions $u=u(x,y)$ of the critical equation ${\displaystyle {\Delta }_{x}u+(\alpha +1){}^2\left|x\right|{}^{2\alpha }{\Delta }_{y}u=-u^{(Q+2)/(Q-2)}\hspace{1em}\text{in}{ℝ}^n={ℝ}^m\times {ℝ}^k,}$ where $(x,y)\in {ℝ}^m\times {ℝ}^k$, $\alpha >0$, and $Q=m+k(\alpha +1)$. In the first part of the article, exploiting the invariance of the equation with respect to a suitable conformal inversion, we prove a “spherical symmetry result for solutions”. In the second part, we show how to reduce the dimension of the problem using a hyperbolic symmetry argument. Given any positive solution $u$ of (1), after a suitable scaling and a translation in the variable $y$, the function $v(x)=u(x,0)$ satisfies the equation ${\displaystyle \mathrm{div}{}_x(p{\nabla }_{x}v)-qv=-p{v}^{(Q+2)/(Q-2)},\hspace{1em}\left|x\right|<1,}$ with a mixed boundary condition. Here, $p$ and $q$ are appropriate radial functions. In the last part, we prove that if $m=k=1$, the solution of (2) is unique and that for $m\ge 3$ and $k=1$, problem (2) has a unique solution in the class of $x$-radial functions