Isoperimetric Reformulation

• Isoperimetric Reformulation is a method that recasts classical perimeter optimization problems into variational, geometric, and analytic frameworks using symmetry and curvature techniques.
• It simplifies high-dimensional issues by applying spherical symmetrization to reduce problems to one-dimensional variational forms governed by Euler–Lagrange ODEs.
• This framework enables the derivation of sharp isoperimetric inequalities and quantitative comparisons for log-convex measures in weighted and curved spaces.

The isoperimetric reformulation refers to the rigorous recasting of the classical isoperimetric problem—minimizing surface area or perimeter for a given measure—within broader geometric, analytic, and variational frameworks. Modern approaches reduce high-dimensional or constrained problems to tractable variational forms, derive necessary ODEs and inequalities, and establish comparison theorems that quantitatively capture the influence of curvature, convexity, and density. In the presence of radially symmetric measures, isoperimetric reformulation leverages symmetrization and variational analysis to systematize and generalize isoperimetric minimization, particularly for log-convex and related measures (Kolesnikov et al., 2010). This framework has deep connections to geometric measure theory, the calculus of variations, and analysis on weighted and curved spaces.

1. Spherical Symmetrization and Reduction to One-Dimensional Variational Problems

For measures with radially symmetric densities on $\mathbb{R}^d$, specifically those with densities $\rho(r) = e^{-v(r)}$, the isoperimetric reformulation is built upon a spherical symmetrization process. This method replaces an arbitrary set $A$ with a symmetrized set $A^*$ which, at each radius $r$, preserves the $(d-1)$-dimensional measure of $A \cap \{|x| = r\}$ as a centered spherical cap or arc. The key property, established via coarea and convexity arguments, is that the $\mu$-surface measure does not increase under this transformation:

$\mu^+(\partial A^*) \leq \mu^+(\partial A)$

This leads to the critical simplification: without loss of generality, isoperimetric candidates can be sought among rotationally symmetric sets. For $d=2$, candidates are of the form

$A = \{(r, \theta) : -f(r) < \theta < f(r)\}$

with $f(r) \in [0, \pi]$, reducing the original high-dimensional minimization to a one-parameter variational problem.

2. Variational Characterization and Euler–Lagrange ODEs

The perimeter and measure of the symmetrized region $A$ admit explicit integral formulas for a given density:

$\mu(A) = 2 \int_{0}^{\infty} r f(r) \rho(r) \, dr$

$$\mu^+(\partial A) = 2 \int_{0}^{\infty} \sqrt{1 + r^2(f'(r))^2} \, \rho(r) \, dr$$

Imposing a fixed measure constraint, the first variation with respect to $f$ leads to an Euler–Lagrange equation for stationary sets. Introducing

$u(r) = \frac{r^2 f'(r)}{\sqrt{1 + r^2(f'(r))^2}}$

yields an ODE:

$u'(r) - v'(r) u(r) = c\, r$

where $c$ is a Lagrange multiplier. This ODE encapsulates the necessary conditions for isoperimetricity and forms the core of the reformulated problem for radially symmetric measures.

3. Qualitative and Numerical Analysis of Solutions

Numerical resolution of the above ODE for families of exponential power-law densities $\rho(r) = C_\alpha e^{-r^{\alpha}}$ with $\alpha \ge 1$ reveals distinct qualitative behaviors:

• For $\alpha > 2$ (super-Gaussian), solutions correspond to non-compact, non-self-intersecting boundaries.
• For $\alpha = 1$ (exponential), solutions are compact, convex, rotationally symmetric, but not Euclidean balls; half-space-like solutions also appear.
• For $1 < \alpha < 2$, a threshold exists distinguishing between compact and non-compact or self-intersecting solutions. Only those symmetric sets with closed, smooth boundaries and appropriate divergence properties of $f'$ at endpoints are genuine isoperimetric minimizers. The optimal shape thus depends sensitively on the growth of $v(r)$, with empirical confirmation via numerical experimentation.

4. Isoperimetric Inequalities and Log-Convex Densities

For log-convex measures ($\mu = e^{V(x)} dx$ with $V$ strictly convex), sharp isoperimetric inequalities are established:

$$\mu^+(\partial A) \geq \frac{1}{\sqrt{1 + \pi^2}} \mu^+(\partial B_r)$$

when $\mu(A) = \mu(B_r)$. In particular, for sufficiently strong convexity, symmetric balls—Euclidean balls centered at the origin—become isoperimetric for large volumes. These inequalities link convexity properties of the potential directly to the isoperimetric profile and provide explicit quantitative lower bounds on the surface measure.

5. Comparison Theorems for General Log-Convex Measures

The reformulation extends to comparison with model measures: if $W''e^{-2W} \ge A^2$, the measure $\mu = e^{W(x)} dx$ can be realized as the image of the model measure

$\nu_A(dx) = \frac{dx}{\cos Ax}$

via a 1-Lipschitz map. Hence,

$I_\mu(t) \geq I_{\nu_A}(t) = e^{At/2} + e^{-At/2}$

This provides concrete comparison theorems relating the isoperimetric profile of a general log-convex measure to that of a model space, echoing analogies to the Bakry–Ledoux theorem for log-concave settings. Similar results for product measures are established, showing that convexity controls the isoperimetric shape quantitatively.

6. Summary of Key Formulas

Context	Formula/type	Role
Euler–Lagrange reformulation	$u'(r) - v'(r) u(r) = c\, r$	Characterizes stationary isoperimetric sets
Perimeter of symmetric set	$$\mu^+(\partial A) = 2 \int \sqrt{1 + r^2 (f'(r))^2} \rho(r) dr$$	Surface measure to minimize
Sharp isoperimetric inequality	$$\mu^+(\partial A) \ge \frac{1}{\sqrt{1 + \pi^2}} \mu^+(\partial B_r)$$	Lower bound for log-convex measures
Comparison with model measure	$\nu_A(dx) = \frac{dx}{\cos A x}$, $I_{\nu_A}(t) = e^{At/2} + e^{-At/2}$	Relates general and model isoperimetric profiles

These forms precisely encapsulate the reformulation’s structure: symmetry reduction, analytic characterization via ODEs, explicit perimeter formulas, and comparison results.

7. Significance and Broader Impact

The isoperimetric reformulation—anchored in symmetrization, variational reduction, and ODE analysis—broadens the classical isoperimetric problem to settings with arbitrary radial densities and log-convexity, offering both empirical classification of minimizers and sharp inequalities linking geometry and potential theory. The approach is foundational for further work in weighted isoperimetry, concentration of measure, stability of geometric inequalities, and analysis on curved and weighted spaces. The combination of continuous (ODE-based) and discrete (comparison-theorem) arguments deepens the interplay between geometry, analysis, and probability, and serves as a prototype for the analysis of isoperimetric phenomena far beyond the Euclidean constant-density case (Kolesnikov et al., 2010).