arXiv subjects

Manwook Han

Publications and source records attributed to Manwook Han.

7 recordsLinked to original sources

arXiv2026

Classical polynomial inequalities for quadratic forms on an octagonal sector

We establish a collection of sharp inequalities for real quadratic forms on the first-quadrant sector of a regular octagon. Starting from the complete extreme-point description of the associated polynomial unit ball, we compute the exact pointwise Bernstein function for the Euclidean gradient. One extreme curve controls the problem: its endpoint is active up to slope $1/2$, after which the maximizer follows an explicit Cardano branch. We obtain the sharp Markov constant $2\sqrt5$, the exact relative quadratic polarization constant $2$, the canonical unconditional constant $3$, and the body-relative Bohr radius $1/\sqrt3$. We also determine the optimal coefficient $\ell_q$-comparison for every $1\le q\le\infty$. The same norm-one polynomial is extremal for all these global constants. At $q=4/3$ the result is a sharp fixed-space coefficient inequality of \textit{Bohnenblust--Hille type}.

math.FA↗

arXiv2026

The unit ball of quadratic forms on an octagonal sector

Let \[\mathfrak O=\{(x,y)\in[0,1]^2:x+y\le \sqrt2\} \] be the first-quadrant sector of a regular octagon. For quadratic forms $P(x,y)=ax^2+bxy+cy^2$, we study the supremum norm over $\mathfrak O$. We obtain a complete five-region formula for the norm, according to whether the norming contact occurs at an endpoint or in the interior of one of the three radial sides. We then prove that the projection of the unit ball onto the $ac$-plane is exactly $[-1,1]^2$, compute both endpoints of every vertical section, and thereby parametrize the entire unit sphere. Finally, we characterize the extreme points of the unit ball as four explicit curves, their negatives, and four pairs of isolated points. The resulting description is fully explicit and reduces subsequent convex extremal problems on this polynomial space to four one-parameter families and finitely many isolated polynomials.

math.FA↗

arXiv2026

Projective norm-attainments and their implications

We show that nuclear norm-attaining operators (resp.\ polynomials) are always $w^*$-dense in the space of integral operators (resp.\ polynomials). Besides, the denseness is in norm if the predual space does not contain any isomorphic copy of $\ell_1$. We also show that there are reflexive spaces for which the set of projective norm-attaining elements does not coincide with the whole projective tensor product (which is indeed also reflexive here). Next, we show that if $Y$ is a II-polyhedral space, then every nuclear operator from an arbitrary space $X$ to $Y^*$ attains its nuclear norm. As a consequence, if $X^*$ or $Y^*$ has the approximation property, then the set of norm-attaining operators from $X^*$ to $Y^{**}$ is dense. Finally, we study proximinality results of a natural subspace of the projective tensor product and obtain an application to integral projective norm-attaining tensors which solves a proposed open question.

math.FA↗

arXiv2026

Weak minimizing property on pairs of classical Banach spaces

We investigate the minimum modulus analogue of the weak maximizing property, termed the \emph{weak minimizing property}. We establish that the pairs $(\ell_p, L^p[0, 1])$ for $2 \leq p < \infty$ and $(\ell_s \oplus_q \ell_q, \ell_r \oplus_p \ell_p)$ for $1 < p \leq r\leq s \leq q < \infty$ satisfy the weak minimizing property. Conversely, we prove that the pairs $(\ell_1, \ell_p)$, $(\ell_1, c_0)$, $(\ell_1, \ell_1)$ and $(c_0, \ell_p)$ fail to satisfy the weak minimizing property.

math.FA↗

arXiv2025

Geometry of the space of compact operators endowed with the numerical radius norm

We investigate the space of bounded linear operators on a Banach space equipped with a norm which is equivalent to the operator norm such that the subspace of compact operators is an M-ideal. In particular, we observe that the space of compact operators on $\ell_p$ $(1<p<\infty)$ equipped with the numerical radius norm is an M-ideal whenever the numerical index of $\ell_p$ is not $0$. On the other hand, we show that the space of compact operators on a Banach space containing an isomorphic copy of $\ell_1$ whose numerical index is greater than $1/2$ is not M-ideals. We also study the proximinality, the existence of farthest points and the compact perturbation property for the numerical radius.

math.FA↗

arXiv2024

M-ideals of compact operators and Norm attaining operators

We investigate M-ideals of compact operators and two distinct properties in norm-attaining operator theory related with M-ideals of compact operators called the weak maximizing property and the compact perturbation property. For Banach spaces $X$ and $Y$, it is previously known that if $\mathcal{K}(X,Y)$ is an M-ideal or $(X,Y)$ has the weak maximizing property, then $(X,Y)$ has the adjoint compact perturbation property. We see that their converses are not true, and the condition that $\mathcal{K}(X,Y)$ is an M-ideal does not imply the weak maximizing property, nor vice versa. Nevertheless, we see that all of these are closely related to property $(M)$, and as a consequence, we show that if $\mathcal{K}(\ell_p,Y)$ $(1<p<\infty)$ is an M-ideal, then $(\ell_p,Y)$ has the weak maximizing property. We also prove that $(\ell_1,\ell_1)$ does not have the adjoint compact perturbation property, and neither does $(\ell_1,Y)$ for an infinite dimensional Banach space $Y$ without an isomorphic copy of $\ell_1$ if $Y$ does not have the local diameter 2 property. As a consequence, we show that if $Y$ is an infinite dimensional Banach space such that $\mathcal{L}(\ell_1,Y)$ is an M-ideal, then it has the local diameter 2 property. Furthermore, we also studied various geometric properties of Banach spaces such as the Opial property with moduli of asymptotic uniform smoothness and uniform convexity.

math.FA↗

arXiv2019

Galkin's Lower bound Conjecure for Lagrangian and orthogonal Grassmannians

Let $M$ be a Fano manifold, and $H^\star(M;\mathbb{C})$ be the quantum cohomology ring of $M$ with the quantum product $\star.$ For $\sigma \in H^*(M;\mathbb{C})$, denote by $[\sigma]$ the quantum multiplication operator $\sigma\star$ on $H^*(M;\mathbb{C})$. It was conjectured several years ago \cite{GGI, GI} and has been proved for many Fano manifols \cite{CL1, CH2, LiMiSh, Ke}, including our cases, that the operator $[c_1(M)]$ has a real valued eigenvalue $\delta_0$ which is maximal among eigenvaules of $[c_1(M)]$. Galkin's lower bound conjecture \cite{Ga} states that for a Fano manifold $M,$ $\delta_0\geq \mathrm{dim} \ M +1,$ and the equlity holds if and only if $M$ is the projective space $\mathbb{P}^n.$ In this note, we show that Galkin's lower bound conjecture holds for Lagrangian and orthogonal Grassmannians, modulo some exceptions for the equality.

math.AG↗