No abstract uploaded!

No abstract uploaded!

No abstract uploaded!

In this paper, we introduce several mixed Lp geominimal surface areas for multiple convex bodies for all $−n\neq p \in R$. Our definitions are motivated from an equivalent formula for the mixed p-affine surface area. Some properties, such as the affine invariance, for these mixed Lp geominimal surface areas are proved. Related inequalities, such as, Alexander-Fenchel type inequality, Santal´o style inequality, affine isoperimetric inequalities, and cyclic inequalities are established. Moreover, we also study some properties and inequalities for the i-th mixed Lp geominimal surface areas for two convex bodies.

The estimate of integral points in an n-dimensional polyhedron has many applications in singularity theory, number theory and toric geometry. The third named author formulated a conjecture (i.e., Yau Number Theoretic Conjecture) which gives a sharp polynomial upper estimate on the number of positive integral points in n-dimensional (n  3) real right-angled simplices. The previous results on the conjecture in low dimension cases (n  6) have been proved by using the sharp GLY conjecture. However, it is only valid in low dimension. The Yau Number Theoretic Conjecture for n = 7 has been shown with a completely new method in [22]. In this paper, on the one hand, the similar method has been applied to prove the conjecture for n = 8, but with more meticulous analyses. The main method of proof is summing existing sharp upper bounds for the number of points in seven-dimensional simplex over the cross sections of eight-dimensional simplex. This is a signicant progress since it sheds light on proving the Yau Number Theoretic Conjecture in full generality. On the other hand, we give a new sharper estimate of the Dickman-De Bruijn function (x; y) for 5  y < 23, compared with the result obtained by Ennola.