Kannan–Lovász–Simonovits conjecture

Choose a point at random in a square and measure the sum of its two coordinates, divided by the square root of two. After scaling each coordinate to have mean zero and variance one, this measurement also has variance one: its average squared deviation from its mean is one. Its squared local rate of change is one too. Can one fixed factor compare these two quantities for every allowed measurement and distribution, even with arbitrarily many coordinates? The distributions are log-concave: their log density is concave wherever positive, as for uniform sampling in a convex shape. They are isotropic, meaning centered with unit variance in every unit direction. Measurements must be locally Lipschitz, with locally bounded rates of change and finite average squared gradient.[1] [2]

If a measurement changes slowly near most sampled points, must its total fluctuation stay comparably small without a dimension penalty? For independent uniform coordinates on $[-\sqrt3,\sqrt3]$, set $f(X)=n^{-1/2}\sum_{i=1}^nX_i$. Independence gives $\operatorname{Var}(f)=1$, while $|\nabla f|^2=1$. The conjecture extends this comparison to every locally Lipschitz measurement with finite gradient energy, not just linear sums. For every dimension and log-concave density $\mu$ on $\mathbb R^n$, assume $\mathbb E_\mu X=0$ and $\operatorname{Cov}_\mu(X)=I$. The Poincaré constant is the least admissible factor in $\operatorname{Var}_\mu(f)\le C_P(\mu)\mathbb E_\mu|\nabla f|^2$. Is $C_P(\mu)\le C$ for one absolute constant independent of both dimension and density?[1] [2]

The linear measurement in the square has the desired comparison, but checking one measurement does not control every locally Lipschitz function. The July 2026 preprint claims a dimension-independent factor of two for quadratic forms and a logarithmic bound for the related geometric constant. Quadratic forms are only one class of measurements, and the general estimate still grows with dimension. Neither claim supplies the universal constant for all measurements in the stated class.[2]

The cited July preprint claims $\psi_n\le C\log^{1/4}n$ for the isoperimetric constant, and $\operatorname{Var}_\mu(f)\le2\mathbb E_\mu|\nabla f|^2$ when $f$ is a quadratic form. The two constants are related through $\psi_\mu^2$, not by identifying $\psi_\mu$ with $C_P(\mu)$. The unrestricted estimate consequently remains dimension dependent. The linear example and the claimed quadratic theorem do not establish the bound for all locally Lipschitz functions.[2]

A probability distribution can have two substantial regions separated by a narrow bottleneck. A measurement that changes across that bottleneck may vary greatly while its average local change stays small. The variance inequality is therefore connected to how cheaply a convex probability distribution can be split into two pieces, and to how far a slowly changing measurement can stray from its mean. This connects local sensitivity, global fluctuations and high-dimensional geometry. Such bottlenecks also enter random-walk sampling analyses; a universal inequality would be geometric input to those analyses, not a complete running-time guarantee.[1] [2]

For these laws, Poincaré, isoperimetric and exponential-concentration estimates are quantitatively connected. Variance measures a global spread, gradient energy measures average local change, and isoperimetry measures the boundary cost of separating probability mass. Their comparison says that a convex density cannot conceal an arbitrarily severe bottleneck after covariance normalization. Controlling the same bottleneck is relevant to the movement of random walks used for sampling. The conjecture would unify these geometric and probabilistic controls across dimension; it does not by itself specify an algorithm or remove its other assumptions.[1] [2]