AI 中文总结
该研究引入凸体协变差函数的光滑径向近似，通过径向(-p)均值构造收敛于经典协变差函数的光滑函数，分析p→∞且水平亏缺δ=p^α（α<0）时的极限，揭示不同α下的过渡体及临界情况的Orlicz投影体框架。
AI 中文摘要
我们引入凸体协变差函数的光滑径向近似。该构造用K和K+x的径向(-p)均值替代交集K∩(K+x)，生成一族函数g_{K,p}，当p→∞时，该族函数逐点收敛于经典协变差函数。对每个固定的p>0，函数g_{K,p}在原点附近光滑，其海森矩阵可根据定义极L₂投影体的算子显式计算。我们研究p→∞且水平亏缺δ=p^α（α<0）的同步极限，归一化水平集呈现三种状态：当α>-1时出现极投影体，当α<-1时出现极L₂投影体，当α=-1时出现临界过渡体。该过渡体由显式的logcosh型边界积分描述，自然将临界状态纳入Orlicz投影体的框架中。
英文摘要
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 \(p\to\infty\). 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 \(p\to\infty\) and the level deficit is \(δ=p^α\), \(α<0\). The normalized level sets exhibit three regimes: the polar projection body appears for \(α>-1\), the polar \(L_2\)-projection body appears for \(α<-1\), and a critical transition body appears at \(α=-1\). 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.
