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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Stability in the isoperimetric problem for convex or nearly spherical domains in $\textbf {R}^ n$
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by Bent Fuglede PDF
Trans. Amer. Math. Soc. 314 (1989), 619-638 Request permission

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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Additional Information
  • © Copyright 1989 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 314 (1989), 619-638
  • MSC: Primary 52A40
  • DOI: https://doi.org/10.1090/S0002-9947-1989-0942426-3
  • MathSciNet review: 942426