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In this note, we provide an adaptation of the Kohler-Jobin rearrangement technique to the setting of the Gauss space. As a result, we prove the Gaussian analogue of the Kohler-Jobin resolution of a conjecture of Pólya-Szegö: when the Gaussian torsional rigidity of a domain is fixed, the Gaussian principal frequency is minimized for the half-space. At the core of this rearrangement technique is the idea of considering a “modified” torsional rigidity, with respect to a given function, and rearranging its layers to half-spaces, in a particular way; the Rayleigh quotient decreases with this procedure. We emphasize that the analogy of the Gaussian case with the Lebesgue case is not to be expected here, as in addition to some soft symmetrization ideas, the argument relies on the properties of some special functions; the fact that this analogy does hold is somewhat of a miracle. 

We formulate a plausible conjecture for the optimal Ehrhard-type inequality for convex symmetric sets with respect to the Gaussian measure. Namely, letting $J_{k-<\#comment/>1}\left(s\right)={\int <\#comment/>}_0^s{t}^{k-<\#comment/>1}{e}^{-<\#comment/>\frac{t^2}{2}}dt$and $c_{k-<\#comment/>1}=J_{k-<\#comment/>1}(+\mathrm{\infty <\#comment/>})$, we conjecture that the function $F:[0,1]\to <\#comment/>\mathbb{R}$, given by $F\left(a\right)=\underset{k=1}{\overset{n}{\sum <\#comment/>}}{1}_{a\in <\#comment/>E_k}\cdot <\#comment/>\left({\mathrm{\beta <\#comment/>}}_k{J}_{k-<\#comment/>1}^{-<\#comment/>1}\right(c_{k-<\#comment/>1}a)+{\mathrm{\alpha <\#comment/>}}_k)$(with an appropriate choice of a decomposition $[0,1]={\cup <\#comment/>}_i{E}_i$and coefficients ${\mathrm{\alpha <\#comment/>}}_i,{\mathrm{\beta <\#comment/>}}_i$) satisfies, for all symmetric convex sets $K$and $L$, and any $\mathrm{\lambda <\#comment/>}\in <\#comment/>[0,1]$, $F\left(\mathrm{\gamma <\#comment/>}\right(\mathrm{\lambda <\#comment/>}K+(1-<\#comment/>\mathrm{\lambda <\#comment/>})L\left)\right)\ge <\#comment/>\mathrm{\lambda <\#comment/>}F\left(\mathrm{\gamma <\#comment/>}\right(K\left)\right)+(1-<\#comment/>\mathrm{\lambda <\#comment/>})F\left(\mathrm{\gamma <\#comment/>}\right(L\left)\right).$We explain that this conjecture is “the most optimistic possible”, and is equivalent to the fact that for any symmetric convex set $K$, itsGaussian concavity power $p_s(K,\mathrm{\gamma <\#comment/>})$is greater than or equal to $p_s(R{B}_2^k\times <\#comment/>{\mathbb{R}}^{n-<\#comment/>k},\mathrm{\gamma <\#comment/>})$, for some $k\in <\#comment/>\{1,\dots <\#comment/>,n\}$. We call the sets $R{B}_2^k\times <\#comment/>{\mathbb{R}}^{n-<\#comment/>k}$round $k$-cylinders; they also appear as the conjectured Gaussian isoperimetric minimizers for symmetric sets, see Heilman [Amer. J. Math. 143 (2021), pp. 53–94]. In this manuscript, we make progress towards this question, and show that for any symmetric convex set $K$in ${\mathbb{R}}^n$, $p_s(K,\mathrm{\gamma <\#comment/>})\ge <\#comment/>\underset{F\in <\#comment/>L^2(K,\mathrm{\gamma <\#comment/>})\cap <\#comment/>Lip\left(K\right):\int <\#comment/>F=1}{sup}\left(2T_{\mathrm{\gamma <\#comment/>}}^F\right(K)-<\#comment/>Var(F\left)\right)+\frac{1}{n-<\#comment/>\mathbb{E}{X}^2},$where $T_{\mathrm{\gamma <\#comment/>}}^F\left(K\right)$is the $F-<\#comment/>$torsional rigidity of $K$with respect to the Gaussian measure.Moreover, the equality holds if and only if $K=R{B}_2^k\times <\#comment/>{\mathbb{R}}^{n-<\#comment/>k}$for some $R>0$and $k=1,\dots <\#comment/>,n$.As a consequence, we get $p_s(K,\mathrm{\gamma <\#comment/>})\ge <\#comment/>Q(\mathbb{E}|X{|}^2,\mathbb{E}\Vert <\#comment/>X{\Vert <\#comment/>}_K^4,\mathbb{E}\Vert <\#comment/>X{\Vert <\#comment/>}_K^2,r(K\left)\right),$where $Q$is a certain rational function of degree $2$, the expectation is taken with respect to the restriction of the Gaussian measure onto $K$, $\Vert <\#comment/>\cdot <\#comment/>{\Vert <\#comment/>}_K$is the Minkowski functional of $K$, and $r\left(K\right)$is the in-radius of $K$. The result follows via a combination of some novel estimates, the $L2$method (previously studied by several authors, notably Kolesnikov and Milman [J. Geom. Anal. 27 (2017), pp. 1680–1702; Amer. J. Math. 140 (2018), pp. 1147–1185;Geometric aspects of functional analysis, Springer, Cham, 2017; Mem. Amer. Math. Soc. 277 (2022), v+78 pp.], Kolesnikov and the author [Adv. Math. 384 (2021), 23 pp.], Hosle, Kolesnikov, and the author [J. Geom. Anal. 31 (2021), pp. 5799–5836], Colesanti [Commun. Contemp. Math. 10 (2008), pp. 765–772], Colesanti, the author, and Marsiglietti [J. Funct. Anal. 273 (2017), pp. 1120–1139], Eskenazis and Moschidis [J. Funct. Anal. 280 (2021), 19 pp.]), and the analysis of the Gaussian torsional rigidity. As an auxiliary result on the way to the equality case characterization, we characterize the equality cases in the “convex set version” of the Brascamp-Lieb inequality, and moreover, obtain a quantitative stability version in the case of the standard Gaussian measure; this may be of independent interest. All the equality case characterizations rely on the careful analysis of the smooth case, the stability versions via trace theory, and local approximation arguments. In addition, we provide a non-sharp estimate for a function $F$whose composition with $\mathrm{\gamma <\#comment/>}\left(K\right)$is concave in the Minkowski sense for all symmetric convex sets. 

We show that for any even log-concave probability measure $\mathrm{\mu <\#comment/>}$on ${\mathbb{R}}^n$, any pair of symmetric convex sets $K$and $L$, and any $\mathrm{\lambda <\#comment/>}\in <\#comment/>[0,1]$, $\mathrm{\mu <\#comment/>}\left(\right(1-<\#comment/>\mathrm{\lambda <\#comment/>})K+\mathrm{\lambda <\#comment/>}L{)}^{c_n}\ge <\#comment/>(1-<\#comment/>\mathrm{\lambda <\#comment/>}\left)\mathrm{\mu <\#comment/>}\right(K{)}^{c_n}+\mathrm{\lambda <\#comment/>}\mathrm{\mu <\#comment/>}(L{)}^{c_n},$where $c_n\ge <\#comment/>n^{-<\#comment/>4-<\#comment/>o\left(1\right)}$. This constitutes progress towards the dimensional Brunn-Minkowski conjecture (see Richard J. Gardner and Artem Zvavitch [Tran. Amer. Math. Soc. 362 (2010), pp. 5333–5353]; Andrea Colesanti, Galyna V. Livshyts, Arnaud Marsiglietti [J. Funct. Anal. 273 (2017), pp. 1120–1139]). Moreover, our bound improves for various special classes of log-concave measures.