Convexity and Applications

这个项目的一个主要重点是高维物体和现象。它们出现在不同的领域，如物理、生物和医学、计算机科学、优化、经济学和材料科学。一个科学或工程问题的数学描述通常需要许多独立的数字，从而产生一个高维的几何空间。例如，如果您要指定一个气体分子在房间中的位置，则需要使用三个数字报告该分子的正面/背面、左右和上下位置。分子运动的方向和速度需要另外三个数字，所以为了描述分子的当前状态，使我们能够根据位置和速度预测它的未来运动，我们总共需要六个独立的数字。如果你想跟踪房间里100个不同的空气分子，那么你需要600个独立的数字坐标来收集所有相关的测量数据。随着这些维度增加了采样的难度，计算也迅速增加，科学家和数学家有时将这种现象称为“维度的诅咒”。然而，随着维度的增加，也会出现一些在低维度中看不到的图案。我们可以利用这些模式，从而利用“维度的诅咒”，使其成为“维度的祝福”。研究这种高维现象是这些项目的目的之一，这些项目是渐近几何分析和仿射凸几何。特别有趣的是高维凸体上的仿射不变泛函。其中最重要的泛函有仿射表面积和p-仿射表面积。它们对应的仿射等周不等式，是由主要研究者和合作者为所有p建立的，比它们的欧几里德同行更强，并与著名的Mahler猜想有关，该猜想在四维和更高维上仍然是开放的。P-仿射表面积与凸体的锥测度的熵直接相关，从而在凸几何和信息论之间建立了联系。这一联系也将在对数凹函数的背景下进一步探索，对数凹函数是凸体在函数域中的自然延伸。此外，仿射表面积自然而然地出现在用多面体逼近凸体的问题中，这是进一步研究的主要课题。目标是建立对近似中涉及的所有相关参数的最优依赖，例如，维度，近似多面体的顶点数量。这些问题将在欧几里得、球面和双曲空间中考虑。这一奖项包括对研究生的支持。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

A major emphasis of this project is on high dimensional objects and phenomena. These appear in areas as diverse as physics, biology and medicine, computer science, optimization, economics, and material sciences. A mathematical description of a scientific or engineering question often requires many independent numbers, leading to a geometric space of high dimension. For example, if you want to specify the location of one gas molecule in a room then you need to report the front/back, side-to-side, and up/down locations of the molecule, using three numbers. The direction and speed of the molecule's motion takes another three numbers, and so to describe enough of the molecule's current state to allow us to predict its future motion from position and velocity we would need six separate numbers in all. If you want to track 100 distinct molecules of the air in the room then you will need 600 independent numerical coordinates to collect all of the relevant measurements. As these dimensions increase the difficulty of sampling, and computation goes up rapidly, a phenomenon scientists and mathematicians sometimes call "the curse of dimensionality." However, there are also patterns that emerge as dimension increases that are not visible in low dimensions. We can exploit those patterns thus taking advantage of the "curse of dimensionality" to make it the "blessing of dimensionality". It is one purpose of these projects to study such high dimensional phenomena.These projects are in asymptotic geometric analysis and affine convex geometry. Of particular interest are the affine invariant functionals on convex bodies in high dimensions. Among the most important such functionals are affine surface area and p-affine surface area. Their corresponding affine isoperimetric inequalities, established by the principal investigator and collaborators for all p, are stronger than their Euclidean counterparts and related to the famous Mahler conjecture which is still open in dimensions four and higher. p-affine surface areas are directly related to entropies of cone measures of convex bodies which establishes a link between convex geometry and information theory. This link will be further explored, also in the context of log concave functions which are a natural extension of convex bodies in the realm of functions. Moreover, affine surface area appears naturally in questions on approximation of convex bodies by polytopes, a further main topic of study. The goal is to establish optimal dependence on all the relevant parameters involved in the approximation, like e.g., dimension, the number of vertices of the approximating polytopes. These issues will be considered in Euclidean, spherical and hyperbolic space. This award includes support for a graduate student.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.