For a convex body $K \subset \mathbb R^d$ let $\Delta(K)$ be the expected distance between two independent uniform points of $K$, and let $\theta(K)$ be the corresponding expectation for normalized surface measure on $\partial K$. The Zaporozhets-Tarasov conjecture asserts $\Delta(K) \le \theta(K)$. We prove this conjecture in case $d=2$. In addition, we give a six-vertex convex polytope in $\mathbb R^3$ for which the reverse strict inequality holds, and obtain counterexamples in every dimension $d \ge 3$ by taking products with segments. Finally, we show that $\theta(K)\ge \frac{\operatorname{per}K}{6}$ for every planar convex body.
For $m\geq 2$, let $c_p(m)$ be the all-dimensional best constant in $$ \left\|\sum_{k=1}^m A_k\right\|_p \leq c_p(m)\left\|\sum_{k=1}^m |A_k|\right\|_p. $$ Tang and Zhang conjectured an explicit formula for every finite $p>1$. We disprove the conjecture with two explicit real $2\times 2$ rank-one matrices at $p=3/2$. The comparison is certified by seven strict rational inequalities and, in particular, places the attained ratio above $207/200$, while the conjectured constant lies below $207/200$. On the positive side, we prove the conjectured sharp bound for every family of rank-at-most-one summands when $2\leq p<\infty$, and classify all equality cases. We also prove the corresponding endpoint statement for $p=\infty$. Finally, for arbitrary complex matrices, we establish the conjectured sharp constant in the case $m=2$, $p=4$.
Let $\mu$ be a log-concave probability measure on $\mathbb R^n$ and let $f\colon\mathbb R^n\to\mathbb R^k$ be a polynomial mapping of degree at most $d$. We show that \[ \mu(f\in A) \le C\bigl(\lambda_k(A)\bigr)^{\frac{1}{k(d-1)+1}} \] for every Borel set $A\subset\mathbb R^k$ whenever the image measure $\mu\circ f^{-1}$ is absolutely continuous. The constant $C$ is independent of the dimension $n$, and the exponent $\frac{1}{k(d-1)+1}$ is sharp. This extends the scalar Carbery--Wright inequality and answers, in the log-concave setting, a question raised by Avni, Glazer, and Larsen. In addition, we show that the density of $\mu\circ f^{-1}$, whenever it exists, belongs to the Nikolskii--Besov space $B^{\frac{1}{k(d-1)+1}}_{1,\infty}(\mathbb R^k)$, with a dimension-free bound for the corresponding norm. A central difficulty in passing from scalar polynomials to vector-valued polynomial mappings is the lack of a suitable nondegeneracy parameter quantifying absolute continuity of $\mu\circ f^{-1}$, as the variance does in the scalar case. Natural candidates such as the covariance matrix or the Jacobian matrix either fail to characterize this property or do not lead to dimension-free estimates. We identify such a parameter and define it to be the covariance matrix of the vector formed by the monomials of degree up to $d^{k-1}$ in the normalized components of $f$. The dimension-free nature of our results allows us to extend Kusuoka's absolute continuity criterion for Gaussian polynomial random vectors to the log-concave setting. Moreover, in this setting, we obtain estimates relating convergence in distribution to convergence in total variation for polynomial random vectors.
Let $S_k(\alpha;K)$ denote the exponential sum over $k$-free integers in the short interval $(N-K,N]$. For $s>0$, we prove essentially tight bounds on the $s$-th moments of $S_k(\alpha;K)$ whenever $K \gg N^{\theta_{k,s}+\epsilon}$ for some $\theta_{k,s}<1/2$. As an immediate consequence, we obtain a lower bound for the $L^1$-mean of the M\"obius-twisted exponential sum over short intervals of length at least $N^{0.49685}$. Moreover, we show that further improvements on all of these results would follow immediately from improvements to an $\ell^2$-estimate involving the M\"obius function.
For a simple graph $G$ of order $n$, let $\lambda_1(G)\ge \cdots \ge \lambda_n(G)$ denote its adjacency eigenvalues. Hong's problem asks for the optimal upper bound for $\lambda_k(G)$. A recent theorem of Sivashankar gives, for every $k\ge3$, \[ \lambda_k(G)\le \frac{(k-2)\sqrt{k+1}+2}{2k(k-1)}\,n-1, \] with sharp examples arising from maximal real equiangular tight frames. In this paper, we characterize the equality case. We also obtain an explicit combinatorial description of the extremal graphs for $\lambda_3$ and $\lambda_4$.
Hitesh Kumar, Bojan Mohar, S. A. Mojallal et al.· 0 citations
Let $\lambda_j(K)$ be the $j$th Dirichlet eigenvalue of a convex body $K$. It is well known that $\lambda_1$ satisfies a Brunn--Minkowski inequality: $K \mapsto \lambda_1(K)^{-1/2}$ is concave on the family of convex bodies. We show that no analogous statement holds for higher eigenvalues. More precisely, for any $j \geq 2$ and $N \geq 2$, if $K \mapsto (f \circ \lambda_j)(K)$ is concave on the family of convex bodies in $\mathbb{R}^N$ for some function $f: (0, \infty) \to \mathbb{R}$, then $f$ must be constant.
Suppose we have $r$ linearly independent hypersurfaces of degree $d$ in $\mathbb{P}^m$ (or $\mathbb{A}^m$) defined over a finite field $\mathbb{F}_q$, whose intersection is at most $k$-dimensional. What is the largest possible number of $\mathbb{F}_q$-rational points in the intersection? We conjecture an exact formula for this problem in both the projective and affine settings, assuming $q\geq d+1$. The case $k=m-1$ recovers the Beelen-Datta-Ghorpade conjecture [2] and the case $k=0$ recovers the zero-dimensional conjecture in [15]. Another interesting special case of the conjecture is $k=m-r$, which corresponds to the complete intersection of $r$ degree $d$ polynomials. The case $r=1$, $k=m-1$ was proven by Serre in [16] who showed that if $F$ is a degree $d$ homogeneous polynomial, then $|V(F)(\mathbb{F}_q)|\leq dq^{m-1}+\pi_{m-2}(q)$. We prove the case $r=2$ and $k=m-2$, that is, if $F_1, F_2$ are coprime, degree $d$ homogeneous polynomials, then $|V(F_1,F_2)(\mathbb{F}_q)|\leq d^2q^{m-2}+\pi_{m-3}(q)$.