Geodesic
Refinement of Isoperimetric Inequality of Minkowski with the Account of Singularities in Boundaries of Intrinsic Parallel Bodies
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 10 (2014) no. 3, pp. 309-319 Cet article a éte moissonné depuis la source Math-Net.Ru

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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}, $B_{A_i}$ 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 $\int\limits_{0}^{q} g(t) df(t) = \lim\limits_{T\to q} \int\limits_{0}^{T} g(t) df(t)$. 
@article{JMAG_2014_10_3_a2,
     author = {V. I. Diskant},
     title = {Refinement of {Isoperimetric} {Inequality} of {Minkowski} with the {Account} of {Singularities} in {Boundaries} of {Intrinsic} {Parallel} {Bodies}},
     journal = {\v{Z}urnal matemati\v{c}eskoj fiziki, analiza, geometrii},
     pages = {309--319},
     year = {2014},
     volume = {10},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JMAG_2014_10_3_a2/}
}
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V. I. Diskant. Refinement of Isoperimetric Inequality of Minkowski with the Account of Singularities in Boundaries of Intrinsic Parallel Bodies. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 10 (2014) no. 3, pp. 309-319. http://geodesic.mathdoc.fr/item/JMAG_2014_10_3_a2/

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