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

V. I. Diskant

Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 10 (2003) no. 1, pp. 40-48

Résumé

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 $A_{-q}(B)$.