Refinement of Isoperimetric Inequality of Minkowski with the Account of Singularities in Boundaries of Intrinsic Parallel Bodies

The following inequalities are proved:
\begin{eqnarray*} S^n(A,B)\geq n^n\sum\limits_{i=0}^{k-1} V(B_{A_i})\left( V^{n-1}(A_i) - V^{n-1}(A_{i+1}) \right) +S^n(A_{-T}(B),B), \end{eqnarray*}

\begin{eqnarray*} S^n(A,B)\geq n^n\int\limits_{0}^{T} g(t) df(t) +S^n(A_{-T}(B),B), \end{eqnarray*}

\begin{eqnarray*} S^n(A,B)\geq n^n\int\limits_{0}^{q} g(t) df(t) +S^n(A_{-q}(B),B), \end{eqnarray*}
where $V(A)$, $V(B)$ stand for the volumes of convex bodies $A$ and $B$ in $\mathbb R^n$ ($n\geq 2$), $S(A,B)$ denotes the area of the surface of the body $A$ relative to the body $B$, $q$ is the capacity factor of the body $B$ with respect to the body $A$, $A_i = A_{-t_i}(B) = A / (t_iB)$ is the inner body parallel to the body $A$ with respect to the body $B$ at a distance $t_i$, $0=t_0 < t_1 <\ldots< t_i< \ldots < t_{k-1}<t_k=T<q$, $B_{A_i}$ is a shape body of $A_i$ relative to $B$, $g(t) = V(B_{A_{-t}(B)})$, $f(t) = - V^{n-1}( A_{-t}(B))$, $\int\limits_{0}^{T} g(t) df(t) $ is the Riemann–Stieltjes integral of the function $g(t)$ by the function $f(t)$, and $\int\limits_{0}^{q} g(t) df(t) = \lim\limits_{T\to q} \int\limits_{0}^{T} g(t) df(t)$.