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We construct an explicit family of real $m\times n$ restricted isometries with $m\to\infty$, $m/n\to 0$, $δ_k\to 0$, and $k=m^{2/3-o(1)}$. We combine Gershgorin's circle theorem with a flat RIP estimate to reduce the problem to choosing a large vertex set in a strongly regular graph with the property that every small subset spans few edges. We select this vertex set using an incidence geometric result due to Zahl. The proof only uses the entrywise absolute value of the Gram matrix despite the conventional wisdom that such proof techniques exhibit a barrier of $m^{1/2}$. We prove that this barrier is actually $m^{2/3}$, and we show that our particular construction exhibits the same barrier.

We introduce Multi-scale Closing Index (MCI), a new dissimilarity measure for evaluating the alignment of two 2D point sets. MCI is bounded between 0 and 1 and is based on multiscale mathematical morphology operations, with smaller values indicating greater structural agreement. The measure was motivated by the need to assess the quality of registration between point sets representing cell populations in biological images. To evaluate its behaviour under controlled conditions, we generated synthetic point clouds from Perlin-noise-based spatial structures, following established approaches for simulating biologically plausible spatial organization. We assessed MCI between those point clouds and different transformed versions, obtained by local perturbation, point removal, translation, rotation and addition of impulsive noise. Across these experiments, the dissimilarity MCI increased gradually and consistently with perturbation severity, while showing a clear discontinuity between clouds originating from the same underlying spatial structure and point clouds generated from different structures. We further compared MCI with commonly used point-set distances, including Hausdorff-based measures and Sliced Wasserstein distance. These experiments show the ability of MCI to capture structural agreement and quantify deviations between point sets, highlighting its potential as a complementary metric for evaluating two-dimensional point-set registration.

We establish an $Ω(d)$ lower bound on the competitive ratio of convex body chasing in $\mathbb{R}^d$ with movement costs measured in any norm. Previously, the best known lower bound valid across all norms was $Ω(d^{1/6})$, established by Papazov (2023); stronger lower bounds were known for $\ell_p$ norms, where Bubeck, Klartag, Lee, Li, and Sellke (2019) had established a lower bound of $Ω(\max\{\sqrt{d}, d^{1-1/p}\})$. In particular, a linear lower bound was known only for the $\ell_\infty$ norm, realized by the standard construction of chasing faces of a hypercube. Our lower bound strictly improves on the prior universal and $\ell_p$ lower bounds for $p < \infty$ and shows that the functional Steiner point algorithm of Sellke (2023) - which has competitive ratio $O(d)$ - is asymptotically optimal for every norm. Our approach is based on the recursive construction of Bubeck, Coester, and Rabani (2023) for metrical service systems, which gives a lower bound of $Ω(r^2)$ for requests of size $O(r)$. We show that requests in their construction, each comprising a set of discrete points in a graph metric, can be mapped to the vertices of convex requests in a $(2r+1)$-dimensional $\ell_p$ vector space while approximately preserving movement costs up to a factor of $O(r)$; the general result then follows via Milman's (1985) quotient-of-a-subspace theorem.