Szymon Wasowicz
The generalized divided differences are introduced. They are applied to investigate some properties characterizing generalized higher-order convexity. Among others some support-type property is proved.
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In this article, we define a new class of convexity called generalized $(h-m)$-convexity, which generalizes $h$-convexity and $m$-convexity on fractal sets $\mathbb{R}^α$ $(0<α\leq 1)$. Some properties of this new class are discussed. Using local fractional integrals and generalized $(h-m)$-convexity, we generalized Hermite-Hadamard (H-H) and Fejér-Hermite-Hadamard (Fejér-H-H) types inequalities. We also obtained a new result of the Fejér-H-H type for the function whose derivative in absolute value is the generalized $(h-m)$-convexity on fractal sets. Some applications to random variables and …
We investigate Sobolev spaces $W^{1,Φ}$ associated to Musielak-Orlicz spaces $L^Φ$. We first present conditions for the boundedness of the Voltera operator in $L^Φ$. Employing this, we provide necessary and sufficient conditions for $W^{1,Φ}$ to contain isomorphic subspaces to $\ell^\infty$ or $\ell^1$. Further we give necessary and sufficient conditions in terms of the function $Φ$ or its complementary function $Φ^*$ for reflexivity, uniform convexity, $B$-convexity and superreflexivity of $W^{1,Φ}$. As corollaries we obtain the corresponding results for Orlicz-Sobolev spaces $W^{1,φ}$ where …
Some new minimax theorems for generalized convexity
本文在非紧致设定下建立了扩展 Simons 极小极大定理等经典结果的双函数极小极大不等式，并证明单函数极小极大等式与 Simons 不等式等价，同时给出了相关应用。
We study the following problem: given n real arguments a1, ..., an and n real weights w1, ..., wn, under what conditions does the inequality w1 f(a1) + w2 f(a2) + ... + wn f(an) >= 0 hold for all functions f with nonnegative kth derivative for some given integer k? Using simple combinatorial techniques, we can prove many generalizations of theorems ranging from the Fuchs inequality to the criterion for Schur convexity.
We consider the geometry of the space of Borel measures endowed with a distance that is defined by generalizing the dynamical formulation of the Wasserstein distance to concave, nonlinear mobilities. We investigate the energy landscape of internal, potential, and interaction energies. For the internal energy, we give an explicit sufficient condition for geodesic convexity which generalizes the condition of McCann. We take an eulerian approach that does not require global information on the geodesics. As by-product, we obtain existence, stability, and contraction results for the semigroup obtai…
In the present paper, we provide several inequalities for the generalized numerical radius of operator matrices as introduced by A. Abu-omar and F. Kittaneh in [3]. We generalize the convexity and the log-convexity results obtained by M. Sababheh in [12] for the case of the numerical radius to the case of the generalized numerical radius. We illustrate our work by providing a positive answer for the question addressed in [12] for the convexity of a certain matrix operator function. Moreover, and motivated by the results of A. Aldalabih and F. Kittaneh in [2] for the case of Hilbert-Schmidt num…
We are concerned about improvements of the modulus of convexity by renormings of a super-reflexive Banach space. Typically optimal results are beyond Pisier's power functions bounds $t^p$, with $p \geq 2$, and they are related to the notion of generalized cotype. We obtain an explicit upper bound for all the modulus of convexity of equivalent renormings and we show that if this bound is equivalent to $t^2$, the best possible, then the space admits a renorming with modulus of power type $2$. We show that a UMD space admits a renormings with modulus of convexity bigger, up to a multiplicative co…
Results on matrix canonical forms are used to give a complete description of the higher rank numerical range of matrices arising from the study of quantum error correction. It is shown that the set can be obtained as the intersection of closed half planes (of complex numbers). As a result, it is always a convex set in $\mathcal C$. Moreover, the higher rank numerical range of a normal matrix is a convex polygon determined by the eigenvalues. These two consequences confirm the conjectures of Choi et al. on the subject. In addition, the results are used to derive a formula for the optimal upper …
Properties of Generalized Polyhedral Convex Multifunctions
本文研究广义多面体凸多值函数在运算下的保持性及最优值函数，给出图像相对内部的表示，并将Bonnans-Shapiro结果推广到局部凸空间。
Towards a Characterization of Matrix Monotonicity via Radial Convexity
该研究针对正定矩阵锥上的矩阵序引入度量的径向凸性，关联其与Loewner矩阵Hadamard平方正定性的关系，在2次情形证明n阶径向凸性等价于n次矩阵单调性，还得到3次情形刻画并讨论推广。
In this paper the theory of uniformly convex metric spaces is developed. These spaces exhibit a generalized convexity of the metric from a fixed point. Using a (nearly) uniform convexity property a simple proof of reflexivity is presented and a weak topology of such spaces is analyzed. This topology called co-convex topology agrees with the usualy weak topology in Banach spaces. An example of a $CAT(0)$-spaces with weak topology which is not Hausdorff is given. This answers questions raised by Monod 2006, Kirk and Panyanak 2008 and Espínola and Fernández-León 2009. In the end existence and uni…
Convexity from order: a theory of faces and generalized separation
本文从序理论视角研究凸集，将面视为强最小值集合，通过定向映射实现广义分离，并定义联合支撑子空间，在仅假设凸性下推广了面运算结果，解决了开放问题。
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