论文标题
在凸面表面积的切片问题的版本上
On a version of the slicing problem for the surface area of convex bodies
论文作者
论文摘要
我们研究表面积而不是体积的切片不等式。这是在尺寸$ n $上是否存在常数$α_n$的问题$ | \ cdot | $表示音量。对于任何固定尺寸，我们都会为这个问题提供负面答案，以及一个较弱的版本，其中部分被投影到超平面上。我们还研究了较低维度和凸体的所有QuermassIntegrals的部分和预测的相同问题。从这些问题开始，我们还引入了许多有关体积和表面积的自然参数，并为它们提供了最佳的上和下限。最后，我们表明，与先前的负面结果相反，问题的变体自然来自同构的busemann等效性的表面积版本 - 切片问题的宠物问题具有肯定的答案。
We study the slicing inequality for the surface area instead of volume. This is the question whether there exists a constant $α_n$ depending (or not) on the dimension $n$ so that $$S(K)\leqα_n|K|^{\frac{1}{n}}\max_{ξ\in S^{n-1}}S(K\capξ^{\perp })$$ where $S$ denotes surface area and $|\cdot |$ denotes volume. For any fixed dimension we provide a negative answer to this question, as well as to a weaker version in which sections are replaced by projections onto hyperplanes. We also study the same problem for sections and projections of lower dimension and for all the quermassintegrals of a convex body. Starting from these questions, we also introduce a number of natural parameters relating volume and surface area, and provide optimal upper and lower bounds for them. Finally, we show that, in contrast to the previous negative results, a variant of the problem which arises naturally from the surface area version of the equivalence of the isomorphic Busemann--Petty problem with the slicing problem has an affirmative answer.
