Jaskaran Singh Kaire | Searcharxiv

arXiv2026

Hadwiger's conjecture for cap bodies

Hadwiger's covering conjecture states that every $n$-dimensional convex body can be covered by at most $2^n$ of its smaller positive homothetic translates, with $2^n$ copies required only for affine images of the $n$-cube. In this note, we confirm Hadwiger's conjecture for the class of cap bodies in all dimensions, bridging recently established cases of $n=3$ and large $n$. The proof uses probabilistic techniques and, for $4\le n \le 15$, computer-assisted linear programming.

math.MG↗

arXiv2026

Illumination number of 3-dimensional cap bodies

The illumination conjecture asserts that any convex body in $n$-dimensional Euclidean space can be illuminated by at most $2^n$ external light sources or parallel beams of light. Despite recent progress on the illumination conjecture, it remains open in general, as well as for specific classes of bodies. Bezdek, Ivanov, and Strachan showed that the conjecture holds for symmetric cap bodies in sufficiently high dimensions. Further, Ivanov and Strachan calculated the illumination number for the class of 3-dimensional centrally symmetric cap bodies to be 6. In this paper, we show that even the broader class of all 3-dimensional cap bodies has the same illumination number 6, in particular, the illumination conjecture holds for this class. The illuminating directions can be taken to be vertices of a regular tetrahedron, together with two special directions depending on the body. The proof is based on probabilistic arguments and integer linear programming.

math.MG↗

arXiv2025

Whitney-type estimates for convex functions

We study Whitney-type estimates for approximation of convex functions in the uniform norm on various convex multivariate domains while paying a particular attention to the dependence of the involved constants on the dimension and the geometry of the domain.

math.CA↗