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About the behavior of isoperimetric difference when turning to parallel body and proving the generalized inequality of Hadwiger
V. I. Diskant Cherkasy State Technological University
Abstract:
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
$A_{-q}(B)$.
MSC: 52A38,
52A40 Received: 17.12.2001
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