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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Stability in the isoperimetric problem for convex or nearly spherical domains in $\textbf {R}^ n$

Author: Bent Fuglede
Journal: Trans. Amer. Math. Soc. 314 (1989), 619-638
MSC: Primary 52A40
MathSciNet review: 942426
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Abstract: For convex bodies $D$ in ${{\mathbf {R}}^n}$ the deviation $d$ from spherical shape is estimated from above in terms of the (dimensionless) isoperimetric deficiency $\Delta$ of $D$ as follows: $d \leq f(\Delta )$ (for $\Delta$ sufficiently small). Here $f$ is an explicit elementary function vanishing continuously at $0$. The estimate is sharp as regards the order of magnitude of $f$. The dimensions $n = 2$ and $3$ present anomalies as to the form of $f$. In the planar case $n = 2$ the result is contained in an inequality due to T. Bonnesen. A qualitative consequence of the present result is that there is stability in the classical isoperimetric problem for convex bodies $D$ in ${{\mathbf {R}}^n}$ in the sense that, as $D$ varies, $d \to 0$ for $\Delta \to 0$. The proof of the estimate $d \leq f(\Delta )$ is based on a related estimate in the case of domains (not necessarily convex) that are supposed a priori to be nearly spherical in a certain sense.

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Keywords: Isoperimetric deficiency, convex bodies, nearly spherical domains, stability
Article copyright: © Copyright 1989 American Mathematical Society