A Smooth Covariogram and the Transition Between Projection Bodies

We introduce a smooth radial approximation of the covariogram of a convex body. The construction replaces the intersection $K\cap (K+x)$ by the radial $(-p)$-mean of $K$ and $K+x$, producing a family $g_{K,p}$ which converges pointwise to the classical covariogram as $p\to \mathrm{\infty }$. For each fixed $p>0$, the function $g_{K,p}$ is smooth near the origin, and its Hessian is computed explicitly in terms of the operator defining the polar $L_2$-projection body. We study the simultaneous limit in which $p\to \mathrm{\infty }$ and the level deficit is $\delta =p^{\alpha }$, $\alpha <0$. The normalized level sets exhibit three regimes: the polar projection body appears for $\alpha >-1$, the polar $L_2$-projection body appears for $\alpha <-1$, and a critical transition body appears at $\alpha =-1$. This transition body is described by an explicit $\log \cosh$-type boundary integral, placing the critical regime naturally within the framework of Orlicz projection bodies.