Related papers: On two recent conjectures in convex geometry
In this short note we provide a new proof of the recent result of Han and Nashimura on the separation of spherical convex sets established in arXiv:2002.06558. Our proof is based on a result stated in locally convex spaces.
In this short note we show, providing counterexamples, that the "two important theorems" in the recent paper [Y, Yuan, Global optimization solutions to a class of non-convex quadratic minimization problems with quadratic constraints, in…
The convex body isoperimetric conjecture in the plane asserts that the least perimeter to enclose given area inside a unit disk is greater than inside any other convex set of area $\pi$. In this note we confirm two cases of the conjecture:…
We refute a recent claim in the literature of a "new" quantum deformation of GL(2).
New cases of the multiplicity conjecture are considered.
It is shown that the weak multidimensional Suita conjecture fails for any bounded non-pseudoconvex domain with $C^{1+\varepsilon_-}$-smooth boundary. On the other hand, it is proved that the weak converse to the Suita conjecture holds for…
This paper has been withdrawn by the author
This paper has been withdrawn by the author, due to an error in the proof of Theorem 3.8.
We present an elementary proof of a conjecture proposed by I. Rasa in 2017 which is an inequality involving Bernstein basis polynomials and convex functions. It was affirmed in positive by A. Komisarski and T. Rajba very recently by the use…
We provide further evidence to favor the fact that rank-one convexity does not imply quasiconvexity for two-component maps in dimension two. We provide an explicit family of maps parametrized by $\tau$, and argue that, for small $\tau$,…
Robin's Conjecture is strengthened, deformed, and proved. Nicolas conjecture follows.
In the paper, the authors prove that the generalized sine function $\sin_{p,q}(x)$ and the generalized hyperbolic sine function $\sinh_{p,q}(x)$ are geometrically concave and geometrically convex, respectively. Consequently, the authors…
We develop the fundamentals of a new theory of convex geometry -- which we call "broken line convex geometry". This is a theory of convexity where the ambient space is the rational tropicalization of a cluster variety, as opposed to an…
This article is motivated by a conjecture proposed by Sinai Robins in 2024. The conjecture asserts that two convex, centrally symmetric sets of positive measure that are not multi-tilers must coincide up to rigid motions if and only if…
Recent developments of affine algebraic geometry, especially the theory of open algebraic surfaces, provide means to systematically explore geometric and topological properties of polynomials in two variables. Nevertheless, there is one…
This is an informal paper presenting historical results around the recent paper of the author about Lang's Conjecture and torsion of elliptic curves. This paper also discusses a few aspects of the proof.
I give an example showing that the recent claimed solution by Feder, Kinne and Rafiey to the CSP Dichotomy Conjecture is not correct.
It is known that the paving conjecture fails for 2-paving projections with constant diagonal 1/2. But the proofs of this fact are existence proofs. We will give concrete examples of these projections and projections with constant diagonal…
Some inequalities for different types of convexity are established.
In this paper, two new lemmas are proved and inequalities are established for co-ordinated convex functions and co-ordinated s-convex functions.