Related papers: On extending the class of convex functions

Employing the notion of operator log-convexity, we study joint concavity$/$ convexity of multivariable operator functions: $(A,B)\mapsto F(A,B)=h\left[ \Phi(f(A))\ \sigma\ \Psi(g(B))\right]$, where $\Phi$ and $\Psi$ are positive linear maps…

The Polyak-{\L}ojasiewicz (P{\L}) inequality extends the favorable optimization properties of strongly convex functions to a broader class of functions. In this paper, we prove a theorem (also obtained by Criscitiello, Rebjock and Boumal in…

For any densely defined, lower semi-continuous trace \tau on a C*-algebra A with mutually commuting C*-subalgebras A_1, A_2, ... A_n, and a convex function f of n variables, we give a short proof of the fact that the function (x_1, x_2,…

First we recall the notion of conxity and log-convexity for real-valued. Then we generalize the trick used by Artin in his famous paper on the Gamma function to find log-convex solutions to the functional equations f(x+1)=g(x)f(x). This…

The matrix convexity and the matrix monotony of a real $C^1$ function $f$ on $(0,\infty)$ are characterized in terms of the conditional negative or positive definiteness of the Loewner matrices associated with $f$, $tf(t)$, and $t^2f(t)$.…

This paper concerns matrix "convex" functions of (free) noncommuting variables, $x = (x_1, \ldots, x_g)$. Helton and McCullough showed that a polynomial in $x$ which is matrix convex is of degree two or less. We prove a more general result:…

Let $\{\hat{P}_{n}(x)\}$ be an orthonormal polynomial sequence and denote by $\{w_{n}(x)\}$ the respective sequence of functions of the second kind. Suppose the Hamburger moment problem for $\{\hat{P}_{n}(x)\}$ is determinate and denote by…

Three results in p-convex geometry are established. First is the analogue of the Levi problem in several complex variables, namely: local p-convexity implies global p-convexity. The second asserts that the support of a minimal p-dimensional…

This paper investigates functions from $\mathbb{R}^d$ to $\mathbb{R} \cup \{\pm \infty\}$ that satisfy axioms of linearity wherever allowed by extended-value arithmetic. They have a nontrivial structure defined inductively on $d$, and…

The article deals with operations defined on convex polyhedra or polyhedral convex functions. Given two convex polyhedra, operations like Minkowski sum, intersection and closed convex hull of the union are considered. Basic operations for…

We prove conditions for the existence of a continuous linear right inverse for a surjective convolution operator in spaces of germs of analytic functions on convex subsets of the complex plane. Considered convex sets have a countable…

An operator convex function on (0,\infty) which satisfies the symmetry condition k(1/x) = x k(x) can be used to define a type of non-commutative multiplication by a positive definite matrix (or its inverse) using the primitive concepts of…

A function is exponentially concave if its exponential is concave. We consider exponentially concave functions on the unit simplex. In a previous paper we showed that gradient maps of exponentially concave functions provide solutions to a…

We study convexity properties of energy functions in plane nonlinear elasticity of incompressible materials and show that rank-one convexity of an objective and isotropic elastic energy $W$ on the special linear group $\mathrm{SL}(2)$…

We denote by ${\rm Hess}^+$ the set of all points $p\in\mathbb{R}^n$ such that the Hessian matrix $H_p(f)$ of the $C^2$-smooth function $f:\mathbb{R}^n\longrightarrow\mathbb{R}$ is positive definite. In this paper we provide a class of…

Gao and Xie (2021) conjectured that the inverse Kazhdan-Lusztig polynomial of any matroid is log-concave. Although the inverse Kazhdan-Lusztig polynomial may not always have only real roots, we conjecture that the Hadamard product of an…

It is shown that the polynomial \[p(t) = \text{Tr}[(A+tB)^m]\] has positive coefficients when $m = 6$ and $A$ and $B$ are any two 3-by-3 complex Hermitian positive definite matrices. This case is the first that is not covered by prior,…

Consider a convex set S defined by a matrix inequality of polynomials or rational functions over a domain. The set S is called semidefinite programming (SDP) representable or just semidefinite representable if it equals the projection of a…

The existence of a homogeneous decomposition for continuous and epi-translation invariant valuations on super-coercive functions is established. Continuous and epi-translation invariant valuations that are epi-homogeneous of degree $n$ are…

We provide three new proofs of the strong concavity of the dual function of some convex optimization problems. For problems with nonlinear constraints, we show that the the assumption of strong convexity of the objective cannot be weakened…