Related papers: On compact sets possessing $q$-convex functions

This note establishes smooth approximation from above for J-plurisubharmonic functions on an almost complex manifold (X,J). The following theorem is proved. Suppose X is J-pseudoconvex, i.e., X admits a smooth strictly J-plurisubharmonic…

We study a properly convex real projective manifold with (possibly empty) compact, strictly convex boundary, and which consists of a compact part plus finitely many convex ends. We extend a theorem of Koszul which asserts that for a compact…

We prove an infinite $(p,q)$-theorem for piercing fat compact convex sets in $\RR^d$ with $k$-flats. Additionally, we develop a new framework through which infinite $(p,q)$-theorems concerning compact sets and $k$-flats can be extended to…

In the spirit of Lelong and Bochner, we show that an upper semi-continuous function defined on a open tube set $\Omega=\omega + i\mathbb{R}^n$ in $\mathbb{C}^n$, where $\omega$ is an open set in $\mathbb{R}^n$, and which is invariant in its…

We give an example of a convex, finite and lower semicontinuous function whose subdifferential is everywhere empty. This is possible since the function is defined on an incomplete normed space. The function serves as a universal…

We studied a new notion of generalized convex functions called $e$-quasi\-con\-ve\-xi\-ty, which encompasses both quasiconvex and $e$-convex functions, including all Lipschitz functions. By extending the standard properties of quasiconvex…

A $q$-rank function is a real-valued function defined on the subspace lattice that is non-negative, upper bounded by the dimension function, non-drecreasing, and satisfies the submodularity law. Each such function corresponds to the rank…

We show that, given a closed convex set $K$ containing the origin in its interior, the support function of the set $\{y\in K^*: \exists x\in K\mbox{ such that } \langle x,y \rangle =1\}$ is the pointwise smallest among all sublinear…

In this article we prove a global result in the spirit of Basener's theorem regarding the relation between q-pseudoconvexity and q-holomorphic convexity: we prove that any smoothly bounded strictly q-pseudoconvex open subset of the complex…

The class of convex sets that admit approximations as Minkowski sum of a compact convex set and a closed convex cone in the Hausdorff distance is introduced. These sets are called approximately Motzkin-decomposable and generalize the notion…

Suppose that $f$ belongs to a suitably defined complete metric space $ {{\cal C}}^{{\alpha}}$ of H\"older $ {\alpha}$-functions defined on $[0,1]$. We are interested in whether one can find large (in the sense of Hausdorff, or lower/upper…

Let $P$ be a finite simplicial comple with underlying space (union of simplices in $P$) $|P|$. Let $Q$ be a subcomplex of $P$. Let $a \geq 0$. Then there exists $K < \infty$, \emph{depending only on $a$ and $Q$,} with the following…

We prove that for a continuum $K\subset \mathbb R^n$ the sum $K^{+n}$ of $n$ copies of $K$ has non-empty interior in $\mathbb R^n$ if and only if $K$ is not flat in the sense that the affine hull of $K$ coincides with $\mathbb R^n$.…

We introduce a wide subclass ${\cal F}(X,\omega)$ of quasi-plurisubharmonic functions in a compact K\"ahler manifold, on which the complex Monge-Amp\`ere operator is well-defined and the convergence theorem is valid. We also prove that…

We provide a specific representation of convex polynomials nonnegative on a convex (not necessarily compact) basic closed semi-algebraic subset K of Rn. Namely, they belong to a specific subset of the quadratic module generated by the…

Replying to three questions posed by N. Shcherbina, we show that a compact psudoconcave set can have the core smaller than itself, that the core of a compact set must be pseudoconcave, and that it can be decomposed into compact…

We use weighted polynomial approximation to prove the existence of a compact set K with non-empty interior and a function f is dense in the space A(K) of all continuous functions on K that are holomorphic in the interior of K, endowed with…

Suppose that a compact quantum group ${\mathcal Q}$ acts faithfully on a smooth, compact, connected manifold $M$, i.e. has a $C^{\ast}$ (co)-action $\alpha$ on $C(M)$, such that $\alpha(C^\infty(M)) \subseteq C^\infty(M, {\mathcal Q})$ and…

Let $A(K)$ be the algebra of continuous functions on a compact set $K\subset\mathbb C$ which are analytic on the interior of $K$, and $R(K)$ the closure (with the uniform convergence on $K$) of the functions that are analytic on a…

A compact subset $K$ of the complex plane $\C$ is a set of polynomial (respectively rational) approximation if $P(K)=A(K)$ (respectively $R(K)=A(K)$), where $P(K)$ (respectively $R(K)$) is the family of functions on $K$ which are uniform…