Abstract
We study an isoperimetric problem described by a functional that consists of the standard Gaussian perimeter and the norm of the barycenter. The second term is in competition with the perimeter, balancing the mass with respect to the origin, and because of that the solution is not always the half-space. We characterize all the minimizers of this functional, when the volume is close to one, by proving that the minimizer is either the half-space or the symmetric strip, depending on the strength of the barycenter term. As a corollary, we obtain that the symmetric strip is the solution of the Gaussian isoperimetric problem among symmetric sets when the volume is close to one. As another corollary we obtain the optimal constant in the quantitative Gaussian isoperimetric inequality.
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Acknowledgements
The first author was supported by INdAM and by the project VATEXMATE. The second author was supported by the Academy of Finland Grant 314227.
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Appendix
Appendix
We first prove the inequalities (4) and (5). In fact, the proof gives us a slightly stronger estimate than (4). We recal that we are assuming \(s \ge 10^3\).
Lemma 5
The following estimates hold:
and
Proof
The right-hand inequality in (72) follows from the isoperimetric inequality \(P_\gamma (D_{\omega ,s}) > P_\gamma (H_{\omega ,s})\) which we may write as \(2 e^{-\frac{a(s)^2}{2}} > e^{-\frac{s^2}{2}}\). This implies
In order to show the right-hand inequality in (73) we first note that the function \(\psi : [0, \infty ) \rightarrow {\mathbb {R}}\),
is increasing. Indeed, its first derivative is
and by a second order analysis it is easy to show that the quantity \(\psi '(x) e^{-\frac{x^2}{2}}\) is positive. The volume condition \(\gamma (D_{\omega ,s}) = \phi (s)\) can be written as
Since \(\psi \) is increasing and \(a(s) > s\) we deduce by the upper bound for a(s) that
Hence we have the right-hand inequality in (73).
To prove the left-hand inequality in (72) we use the above estimate to obtain
In order to prove the inequality we need to show that
This is equivalent to
Use the fact that for \(0<y<1/9\) it holds \(\ln (1 + y) \ge 9y/10\) to estimate
The claim follows from the fact that \(\ln ^2 \left( 2-2/s^2\right) < 9(1 - \ln 2)/5\).
In order prove the left-hand inequality in (73) we first obtain, by integrating by parts twice, that
This implies
Then the volume condition \(\gamma (D_{\omega ,s}) = \phi (s)\) yields
and therefore we have by (72) that
\(\square \)
Finally we prove the perimeter bounds in (10).
Lemma 6
Let E be a minimizer of (7). Then it holds
Proof
The bound from above follows by the minimality and by (73):
The proof of the lower bound is slightly more difficult. Let \({\bar{s}}\) be such that \(\gamma (E) = \phi ({\bar{s}})\). The value \({\bar{s}}\) has to be non-negative, otherwise \({\mathcal {F}}(E)>{\mathcal {F}}({\mathbb {R}}^n\setminus E)\). If \({\bar{s}}\le s\), then the claim follows easily by the Gaussian isoperimetric inequality. If instead \({\bar{s}}> s\), then again by the isoperimetric inequality we have
Define function \(f:[s, \infty ) \rightarrow {\mathbb {R}}\), \(f(x) := e^{-\frac{x^2}{2}} + (s + 1) \int _s^{x} e^{-\frac{t^2}{2}} \, dt\). By differentiating we get
The function is thus increasing up to \(x = s + 1\) and then decreasing. Denote \({\hat{s}}=s+ \frac{1}{6s}\). Let us show that \(f(x) > {\mathcal {F}}(D_{\omega ,s})\) for every \(x \ge {\hat{s}}\).
Note that \(f'(x) \ge \frac{1}{2} e^{-\frac{s^2}{2}}\) for every \(x \in (s, {\hat{s}})\). Therefore since \(f(s) = e^{-\frac{s^2}{2}}\) we get
Moreover we have by (74) that
By the earlier analysis we deduce that for every \(x \ge {\hat{s}}\) it holds
Hence we conclude by (73) that \(f(x) > P_\gamma (D_{\omega ,s})= {\mathcal {F}}(D_{\omega ,s})\) for every \(x \ge {\hat{s}}\). This in turn implies that necessarily \({\bar{s}} < {\hat{s}}\). By the isoperimetric inequality we then have that
\(\square \)
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Barchiesi, M., Julin, V. Symmetry of minimizers of a Gaussian isoperimetric problem. Probab. Theory Relat. Fields 177, 217–256 (2020). https://doi.org/10.1007/s00440-019-00947-9
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DOI: https://doi.org/10.1007/s00440-019-00947-9
Mathematics Subject Classification
- 49Q20
- 60E15