Super-Gaussian directions and the compact-set ambiguity

Definitions and mathematical statement

Grigoris Paouris asks whether every compact set is anti-$\psi_2$ in some direction. As stated this lacks a probability measure, a location/scale normalization, a tail range, and a one-sided versus absolute-tail convention. Arbitrary compact sets may have zero Lebesgue volume. A nontrivial lower tail bound for a compactly supported distribution cannot hold for all unbounded thresholds.

A natural restricted interpretation takes a compact set $K\subseteq{\mathbb{R}}^n$ of positive volume, $X$ uniform on $K$, and $m_\theta={\mathbb{E}}|\langle X,\theta\rangle|$. An $L_1$-normalized absolute-tail formulation would seek universal $c,C>0$ and an admissible threshold range containing $t=1$, such that some unit $\theta$ satisfies \[ \Pr\{|\langle X,\theta\rangle|\geq t m_\theta\}\geq c e^{-Ct^2} \] throughout that range. This is an explicit interpretation, not a recovered definition from the one-line question.

The supplied aid refutes this version even for centered rotationally invariant positive-volume compact sets: for every $n\geq1$ and $0<\epsilon<1/2$ there is such $K$ with \[ \Pr\{|\langle X,\theta\rangle|\geq m_\theta\}\leq\epsilon \qquad\text{for every }\theta\in S^{n-1}. \] An explicit realization of its ball-plus-shell description is the following. Let $a_n=\int_{S^{n-1}}|u_1|\,{\,\mathrm{d}}\sigma(u)>0$ for probability surface measure, choose $R>\max(1,1/(\epsilon a_n))$, set $T=(R^n+\epsilon/(1-\epsilon))^{1/n}$, and take \[ K=B_2^n\ \cup\ \{x:R\leq\|x\|_2\leq T\}. \] The shell has exactly fraction $\epsilon$ of the uniform mass, while $m_\theta\geq\epsilon R a_n>1$ in every direction, so only shell points can contribute at that threshold. This defeats any universal positive lower bound already at $t=1$; the example is not convex.

The aid separately reports Klartag's affirmative theorem for densities with a median-based normalization and an affirmative $L_1$-normalized convex-body version via log-concave scale comparison. Those claims are not contradictory to the nonconvex compact-set example. Their exact median convention, constants, sidedness, and finite threshold range are absent, so no stronger theorem is fabricated from the summary.

Short English statement

The compact-set question is undefined without measure and normalization. Under a natural uniform-measure, first-absolute-moment normalization it is false in every dimension, by concentrating most mass in a ball and a little in a distant shell. The supplied positive median and convex-body variants are distinct statements.

Sources

• AIM workshop Algorithmic convex geometry, problem 8 attributed to Grigoris Paouris: https://aimath.org/WWN/convexgeometry/convexgeometry.pdf.
• The attached aid names Klartag's density theorem and log-concave scale comparison, without full bibliographic entries or exact theorem formulations.
• Local registry data/unsolvedmath/problems.json, permanent ID 20001295, exact source code catalogue entry; canonical location aim-convex-geometry-notes.json notes[26]. Attached attempt 1, counterexample, problem status at run partially solved, rigor proved, low novelty confidence; machine-generated, not peer reviewed, requiring independent verification before citation.