Refinement of Isoperimetric Inequality of Minkowski with the Account of Singularities in Boundaries of Intrinsic Parallel Bodies
V. I. Diskant Cherkasy State Technologic University, 460 Shevchenko Blvd., Cherkasy 18006, Ukraine
Abstract:
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 $\int\limits_{0}^{q} g(t) df(t) =
\lim\limits_{T\to q} \int\limits_{0}^{T} g(t) df(t)$.
Key words and phrases:
convex body, isoperimetric inequality, Minkowski inequality.
MSC: 53B50 Received: 14.05.2013
Revised: 23.12.2013
Language: English
DOI:
10.15407/mag10.03.309
Fulltext:
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