arXiv subjects

Christos Pandis

Publications and source records attributed to Christos Pandis.

6 recordsLinked to original sources

arXiv2026

Variance lower bounds for geometric functionals of rotationally invariant log-concave random polytopes

We establish variance lower bounds for the intrinsic volumes and the face numbers of random polytopes, generated by independent samples from rotationally invariant log-concave probability measures on $\mathbb{R}^d$ with full support. Our results extend the corresponding Gaussian lower bounds of B\'ar\'any and Vu and of B\'ar\'any and Th\"ale to this broader class of distributions. Our proof builds on the geometric construction introduced by B\'ar\'any and Vu, for which we develop an alternative treatment based on barycentric coordinates. We combine it with local variance estimates for both intrinsic volumes and the number of faces.

math.PR↗

arXiv2026

Geometric Bounds for the Mean Gauge and the Mean Width in Isotropic Position

Let $K \subset \mathbb{R}^n$ be an origin-symmetric convex body and assume that its uniform probability measure is isotropic in the probabilistic normalization, namely \[ \int_K x \otimes x \, d\mu_K(x) = \mathrm{Id}_n. \] We give deterministic geometric proofs of \[ M(K) \leq C \frac{\log(n)}{\sqrt{n}} \qquad \text{and} \qquad M^*(K) \leq C \sqrt{n} \, \log(n), \] where \[ M(K) = \int_{\mathbb{S}^{n-1}} \|\theta\|_K \, d\sigma(\theta), \qquad M^*(K) = \int_{\mathbb{S}^{n-1}} h_K(\theta) \, d\sigma(\theta). \] Combining both estimates yields \[ M(K) M^*(K) \leq C \log^2(n). \] The first proof uses a quadratic aggregate of dyadic centroid bodies. The second uses the analogous weighted aggregate of the Laplace bodies $p\{\Lambda_K \leq p\}^{\circ}$, which are equivalent to the centroid bodies by the work of Klartag and E. Milman. In both cases, curvature at each dyadic scale outside a subspace of codimension $O(p)$ leads, via the min--max principle, Legendre duality, and the spherical Laplacian, to the required estimate. The only high-dimensional input is the dimension-free small-ball consequence of the slicing theorem.

math.MG↗

arXiv2026

Restricted Hyperplane Sections of the Cross-Polytope and the Simplex

We give a new proof of Webb's theorem on maximal central hyperplane sections of the regular $n$-simplex $\Delta_n$, viewed in its standard embedding in $\mathbb R^{n+1}$. A similar method also yields sharp maximal estimates for non-central sections of $\Delta^n$ whose distance $d$ from the barycenter is small, namely $d< \sqrt{\frac{1}{(n+1)(2n+1)}}.$ Moreover, we obtain sharp volume estimates for central hyperplane sections of the cross-polytope $B_1^n$ that pass through the barycenter of a facet.

math.MG↗

arXiv2026

Topological Genericity and Large Linear Structures in Function Spaces

We prove that the set of integrable functions on the unit circle for which the analogue of Paley's theorem for $H^1$ fails is residual in $L^1(\mathbb T)$. Moreover, we establish algebraic genericity and spaceability results in several Hardy-type function spaces under prescribed conditions on Taylor coefficients, extending phenomena considered in \cite{pandis2024some,NestoridisGenericH1}.

math.CV↗

arXiv2026

On Extremal Volume Projections of the Simplex and the Cube

Let $\Delta_n$ and $Q_n$ denote the regular $n$-simplex of side length $\sqrt{2}$ embedded in $\mathbb{R}^{n+1}$ and the volume one cube in $\mathbb{R}^n$, respectively. We derive a closed-form formula for the hyperplane volume projections of $\Delta_n$, which also yields the directions achieving the extremal volume. Moreover, we revisit the problem of extremal planar projections of $Q_n$. In addition, we present generalizations within the framework of $L_p$-projection bodies.

math.MG↗

arXiv2025

Sharp inequalities for symmetric polynomials, Hunter's conjecture, and moments of exponential random variables

We prove Hunter's conjecture on complete homogeneous symmetric polynomials. For even $n$ and every integer $k\geq 1$, we show that under the constraint $\sum_{i=1}^n a_i^2=1$ the global minimum of the even-degree polynomial $h_{2k}(a_1,\dots,a_n)$ is attained precisely at the half-plus/half-minus vector and we compute the optimal value in closed form. The proof combines algebraic properties of $h_{2k}$ with the probabilistic representation $k!\,h_k(a)=\mathbb{E}(\sum_{i=1}^n a_iX_i)^k$, where $X_1,\dots,X_n$ are i.i.d. standard exponential random variables with density $e^{-x}1_{x>0}$ and a combinatorial identity. This viewpoint further yields sharp upper and lower bounds for $\mathbb{E}|\sum_{i=1}^n a_iX_i|^{q}$ under natural constraints on the coefficients, including the spherical constraint $\sum a_i^2=1$ combined with the non-negative regime $a_i\ge0$, or the centred regime $\sum a_i=0$. Moreover, we determine the exact minimum of $h_{2k}$ on the $\ell_\infty$-sphere $S_\infty = \{a \in \mathbb{R}^n : \|a\|_\infty = 1\}$, which yields sharp norm comparison inequalities between the matrix norms induced by complete homogeneous symmetric polynomials and the classical operator and Schatten norms.

math.PR↗