Log-Volume Isotropy Metric

• The Log-Volume Isotropy Metric is defined as an affine-invariant scalar that measures the mass concentration of a log-concave function relative to its optimal John ellipsoid.
• It generalizes the classical volume ratio for convex bodies, yielding new stability bounds in affine isoperimetric and Sobolev inequalities.
• Practical examples like Gaussian densities and uniform ball functions show that values near 1 indicate near-isotropy while larger values denote greater mass dispersion.

The Log-Volume Isotropy Metric is an affine-invariant scalar measuring the extent to which the mass of a log-concave function is spread around a distinguished ellipsoidal subset in $\mathbb{R}^n$. For an integrable log-concave function $f:\mathbb{R}^n\to[0,+\infty)$, the metric quantifies isotropy by comparing the total mass of $f$ to the mass captured by its associated John ellipsoid at an optimal level. This concept generalizes the classical volume ratio for convex bodies, leading to new stability statements in functional affine isoperimetric inequalities and connecting with the isotropic position and covariance structure of $f$ (Alonso-Gutiérrez et al., 2015).

1. John’s Ellipsoid for Log-Concave Functions

For an integrable log-concave $f:\mathbb{R}^n\rightarrow[0,+\infty)$, there exists a unique ellipsoidal function

$E_f(x) = t_0\,\|f\|_\infty\,1_{E_f}(x)$

where $E_f = x_0 + A B^n_2$, $t_0\in(0,1]$, $\|f\|_\infty=\max_x f(x)$, $A\in GL(n)$ symmetric positive-definite, and $x_0\in\mathbb{R}^n$ chosen so that:

• $E_f(x)\leq f(x)$ for all $x$,
• $\int_{\mathbb{R}^n} E_f(x)\,dx$ is maximized among all possible such ellipsoidal functions subordinate to $f$.

Operationally, for each $t\in(0,1]$, define the superlevel set $K_t(f) = \{x : f(x) \geq t\,\|f\|_\infty\}$. $E_t(f)$ denotes the maximal-volume ellipsoid inscribed in $K_t(f)$. The optimal $t_0$ maximizes the functional $\varphi_f(t) = t\,\|f\|_\infty\,|E_t(f)|$, leading to assignment $E_f = E_{t_0}(f)$ and height $a_0 = t_0\,\|f\|_\infty$.

2. Definition and Properties of the Integral-Ratio Metric

The Log-Volume Isotropy Metric is formalized as the "integral-ratio" of $f$:

$$I(f) = \left(\frac{\int_{\mathbb{R}^n} f(x)\,dx}{\int_{\mathbb{R}^n} E_f(x)\,dx}\right)^{1/n} = \left(\frac{\max_{0<t\leq 1}\varphi_f(t)}{\varphi_f(t_0)}\right)^{1/n}$$

Key features include:

• $I(f)\geq 1$, with equality if and only if all $E_t(f)$ have constant volume (i.e., $f$ is an exponential supported on an ellipsoid).
• Affine invariance: $I(f)$ remains unchanged under invertible affine transformations.
• For indicator functions $f=1_K$, $I(f)$ specializes to the classical volume-ratio $(|K|/|E(K)|)^{1/n}$, where $E(K)$ is the John ellipsoid of the convex body $K$.

Geometrically, $I(f)$ measures how much of the total mass of $f$ is concentrated on its optimal inscribed ellipsoid $E_f$. Low values of $I(f)$ (close to 1) indicate $f$ is nearly "volume-isotropic," with most mass situated on $E_f$.

3. Relations to Covariance and Functional Isotropic Position

Let $M=\int_{\mathbb{R}^n}f(x)\,dx$, mean $\mu=\frac{1}{M}\int x f(x)\,dx$, and covariance $Cov(f)=\frac{1}{M}\int (x-\mu)(x-\mu)^T f(x)\,dx$. The isotropic position for $f$ is defined by $\mu=0$ and $Cov(f)=L_f^2\,I_n$ for some $L_f>0$.

By applying an affine transformation $g(y)=f(x_0 + A y)$ such that the John ellipsoid of $g$ is $B^n_2$, one gets $Cov(g)=A^{-1}Cov(f)(A^{-1})^T=L_f^2\,I_n$, which yields $A=c L_f I_n$ for $c>0$. In this normalization, it follows that

$$I(f) = \frac{1}{t_0\, (\det A)^{1/n}} = \frac{1}{t_0\,c\,L_f}$$

This establishes control relations between $I(f)$, the isotropic constant $L_f$, and the optimal threshold $t_0$.

4. Extremal Cases and Model Computations

Characteristic model cases elucidate the metric's behavior:

• Gaussian density: For $f_G(x)=(2\pi)^{-n/2}e^{-\frac{|x|^2}{2}}$,
  • Level-sets are balls of radius $r(t)=\sqrt{-2\log(t(2\pi)^{n/2})}$,
  • Maximizer $r=\sqrt{n}$ yields $t_0=(2\pi)^{-n/2}e^{-n/2}$, $E_f=B^n_2(\sqrt{n})$,
  • The mass ratio becomes $$I(f_G)=\left(\frac{\Gamma(n/2)}{\gamma(n/2,n/2)}\right)^{1/n}$$.
• Uniform on the unit ball: For $f(x)=1/|B^n_2|$ on $|x|\leq 1$,
  • $K_t(f)$ is always the unit ball and $E_f$ matches it,
  • $I(f)=1$.

These examples illustrate that $I(f)$ attains its minimum in the uniform case, while the Gaussian demonstrates near-optimal spread consistent with classical isotropy, with $I(f_G) \sim (2\pi e/n)^{1/2}$ in high dimensions.

5. Connections to Functional Affine Isoperimetric Inequality

Define the polar-projection body $II^*(f)$ by the norm

$$\|x\|_{II^*(f)} = 2\int_0^{\infty} |P_{x^\perp}K_t(f)|\, dt = \int_{\mathbb{R}^n}|\partial_x u(y)|\, dy$$

where $u = -\log f$ and $P_{x^\perp}$ denotes geometric projection. For $f\in W^{1,1}(\mathbb{R}^n)$ log-concave, the following reverse functional Petty/Sobolev inequality holds:

$$\|f\|_{n/(n-1)}\, |II^*(f)|^{1/n} \geq e^{\frac{1}{M}\int f\log f - \log \|f\|_\infty}\, I(f)^{-1}$$

where $M=\int f$. This theorem asserts that a lower bound on $I(f)$ prevents collapse of the left-hand side in affine Sobolev inequalities, integrating geometric and analytic structure via $I(f)$.

6. Interpretations and Significance

The Log-Volume Isotropy Metric $I(f)$ provides a quantitative measure of the "even spread" of a log-concave function with respect to its John ellipsoid. Small values (close to 1) signal near-isotropic log-concave functions, as nearly all mass is located within a maximally inscribed ellipsoid. Larger values indicate deviations from isotropy, with mass distribution extending beyond a single ellipsoidal core. $I(f)$ enters stability estimates for functional affine inequalities and controls the isotropic constant $L_f$ via $I(f)\approx 1/(t_0 L_f)$.

A plausible implication is that $I(f)$ acts as a functional proxy for verifying or enforcing isotropic conditions in high-dimensional analysis, convex geometry, and related stability questions. This suggests potential utility in contexts where traditional volume ratio methods are inadmissible or insufficient for log-concave densities (Alonso-Gutiérrez et al., 2015).