Mixed Volumes, the Wigner Caustic, and Isoperimetric Inequalities

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.