A Hyperplane Inequality for Measures of Convex Bodies in ℝn, n≤4
A Hyperplane Inequality for Measures of Convex Bodies in ℝn, n≤4
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ℝn, n≤4 中凸体测度的超平面不等式
DOI：
10.1007/s00454-011-9362-8
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发表时间：
2012
影响因子：
0.8
通讯作者：
A. Koldobsky
中科院分区：
文献类型：
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作者：
A. Koldobsky
摘要设2≤n≤4。我们证明，对于 ℝn 中具有均匀连续密度的任意测度 μ 和 ℝn 中任何原点对称凸体 K，
$$\mu(K) \le\frac{n}{n-1}\frac{|B_2^n|^{\frac{n-1}{n}}}{|B_2^{n-1}|}\max_{\xi\in S^{n-1}} \mu\bigl(K\cap\xi^\bot\bigr)\operatorname{Vol}_n(K)^{1/n},$$ 其中 xi⊥ 是中心ℝn 中垂直于 ϵ 的超平面，$|B_{2}^{n}|$ 是 ℝn 中单位欧氏球的体积。这种不等式是尖锐的，它将维度高达四的超平面不等式推广到任意测量值的设置来代替体积。为了证明这个不等式，我们首先在任意测量的 Busemann-Petty 问题的肯定情况下建立稳定性，在以下意义上：如果 ε>0，K 和 L 是 ℝn、n≤4 中的原点对称凸体，并且
$$\mu\bigl(K\cap\xi^\bot\bigr) \le\mu\bigl(L\cap\xi^\bot\bigr) +\varepsilon,\quad \forall\xi\in S^{n-1},$$ 那么
$$\mu(K)\le\mu(L) + \frac{n}{n-1}\frac{|B_2^n|^{\frac {n-1}{n}}}{|B_2^{n-1}|} \operatorname{Vol}_n(K)^{1/n}\varepsilon。$$
AbstractLet 2≤n≤4. We show that for an arbitrary measure μ with even continuous density in ℝn and any origin-symmetric convex body K in ℝn,
$$\mu(K) \le\frac{n}{n-1}\frac{|B_2^n|^{\frac{n-1}{n}}}{|B_2^{n-1}|}\max_{\xi\in S^{n-1}} \mu\bigl(K\cap\xi^\bot\bigr)\operatorname{Vol}_n(K)^{1/n},$$ where ξ⊥ is the central hyperplane in ℝn perpendicular to ξ, and $|B_{2}^{n}|$ is the volume of the unit Euclidean ball in ℝn. This inequality is sharp, and it generalizes the hyperplane inequality in dimensions up to four to the setting of arbitrary measures in place of volume. In order to prove this inequality, we first establish stability in the affirmative case of the Busemann–Petty problem for arbitrary measures in the following sense: if ε>0, K and L are origin-symmetric convex bodies in ℝn, n≤4, and
$$\mu\bigl(K\cap\xi^\bot\bigr) \le\mu\bigl(L\cap\xi^\bot\bigr) +\varepsilon,\quad \forall\xi\in S^{n-1},$$ then
$$\mu(K)\le\mu(L) + \frac{n}{n-1}\frac{|B_2^n|^{\frac {n-1}{n}}}{|B_2^{n-1}|} \operatorname{Vol}_n(K)^{1/n}\varepsilon.$$