混合体积、维格纳焦散与等周不等式
Mixed Volumes, the Wigner Caustic, and Isoperimetric Inequalities
AI总结：
该研究通过支撑函数奇偶分解推导混合体积相关恒等式，得到强化等周不等式，给出低维显式公式，证明投影转移界及任意维常宽体展开式。
AI中文摘要：
设$K$是$\boldsymbol{R}^n$中的凸体，$C=\frac12(K+(-K))$为其中心对称化体。我们通过支撑函数的奇偶分解研究缺陷$\boldsymbol{\tau}_k(K)=W_{n-k}(C)-W_{n-k}(K)$，推导精确的偶混合体积展开式，将其克拉夫楚克变换与混合差体系数对应，得到库博塔公式，表明各缺陷为投影对称化增益的平均值。这些恒等式导出强化的等周不等式：二次缺陷是投影维格纳焦散的绝对定向面积的平均值，服从精确的球谐稳定性估计。我们给出2、3、4维的显式公式，4维展开含符号不定的4次维格纳体积；在三次谐函数平面上，找到使该体积消失的非球面常宽体。最后，证明投影转移界及任意维常宽体的展开式。
英文摘要：
Let $K\subset\mathbb{R}^n$ be a convex body and $C=\frac12(K+(-K))$ its central symmetral. We study the defects $\mathcal A_k(K)=W_{n-k}(C)-W_{n-k}(K)$ through the even--odd decomposition of the support function. We derive exact even mixed-volume expansions, identify their Krawtchouk transform with the mixed difference-body coefficients, and obtain a Kubota formula expressing each defect as the averaged symmetrization gain of projections. These identities yield strengthened isoperimetric inequalities. The quadratic defect is the average of the absolute oriented areas of projected Wigner caustics and obeys a sharp spherical-harmonic stability estimate. We give explicit formulas in dimensions $2$, $3$, and $4$. The four-dimensional expansion includes a sign-indefinite quartic Wigner volume; on a plane of cubic harmonics we find nonspherical constant-width bodies for which it vanishes. Finally, we prove a projection-transfer bound and an arbitrary-dimensional expansion for constant-width bodies.
