About the behavior of isoperimetric difference when turning to parallel body and proving the generalized inequality of Hadwiger

The following inequalities are proved:
\begin{gather*} V_1^n(A,B)-V(B)V^{n-1}(A)\ge V_1^n(A_{-p}(B),B)-V(B)V^{n-1}(A_{-p}(B)), \\ V_1^n(A,B)-V(B_A)V^{n-1}(A)\ge V_1^n(A_{-p}(B),B)-V(B_A)V^{n-1}(A_{-p}(B)), \\ S^n(A,B)\ge n^n V(B_A)V^{n-1}(A)+S^n(A_{-q}(B),B), \end{gather*}
in which $V(A)$, $V(B)$ — the volumes of convex bodies $A$ and $B$ in $R^n$ ($n\ge 2$), $V_1(A,B)$ — first mixed volume bodies $A$ and $B$, $S(A,B)=nV_1(A,B)$, $q$ — coefficient of capacity $B$ in $A$, $p\in [0,q]$, $A_{-p}(B)$ — internal body which is to parallel to body $A$ relatively to $B$ on the distance $p$, $B_A$ — form-body of body $A$ relatively to $B$. The left part of the first inequality is the isoperimetric difference of $A$ relatively to $B$. The first inequality confirms that when turning from $A$ to $A_{-p}(B)$ the isoperimetric difference relatively to $B$ does not increase. The second inequality proves the first one taking into account the peculiarities on the border of body $A$ relatively to $B$. The third inequality proves the generalization of the inequality of Hadwiger [4] taking into account the degeneracy of $A_{-q}(B)$.