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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.
Received: 14.05.2013 Revised: 23.12.2013
Citation:
V. I. Diskant, “Refinement of Isoperimetric Inequality of Minkowski with the Account of Singularities in Boundaries of Intrinsic Parallel Bodies”, Zh. Mat. Fiz. Anal. Geom., 10:3 (2014), 309–319
Linking options:
https://www.mathnet.ru/eng/jmag596 https://www.mathnet.ru/eng/jmag/v10/i3/p309
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Abstract page: | 126 | Full-text PDF : | 48 | References: | 33 |
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