On the nontrivial projection problem (2008)
Szarek, Stanislaw J., Tomczak-Jaegermann, Nicole
The Nontrivial Projection Problem asks whether every finite-dimensional normed space of dimension greater than one admits a well-bounded projection of non-trivial rank and corank or, equivalently,...
Szarek, Stanislaw J., Werner, Elisabeth, Zyczkowski, Karol
We investigate the set a) of positive, trace preserving maps acting on density matrices of size N, and a sequence of its nested subsets: the sets of maps which are b) decomposable, c) completely...
Still more on norms of completely positive maps (2006)
King and Ruskai asked whether the norm of a completely positive map acting between Schatten classes of operators is equal to that of its restriction to the real subspace of self-adjoint operators....
Tensor products of convex sets and the volume of separable states on N qudits (2005)
Aubrun, Guillaume, Szarek, Stanislaw J.
This note deals with estimating the volume of the set of separable mixed quantum states when the dimension of the state space grows to infinity. This has been studied recently for qubits; here we...
Saturating Constructions for Normed Spaces (2004)
Szarek, Stanislaw J., Tomczak-Jaegermann, Nicole
We prove several results of the following type: given finite dimensional normed space V there exists another space X with log (dim X) = O(log (dim V)) and such that every subspace (or quotient) of X,...
Saturating Constructions for Normed Spaces II (2004)
Szarek, Stanislaw J., Tomczak-Jaegermann, Nicole
We prove several results of the following type: given finite dimensional normed space V possessing certain geometric property there exists another space X having the same property and such that (1)...
Metric Entropy of Homogeneous Spaces (1997)
For a (compact) subset $K$ of a metric space and $\varepsilon > 0$, the {\em covering number} $N(K , \varepsilon )$ is defined as the smallest number of balls of radius $\varepsilon$ whose union...
Metric entropy of homogeneous spaces and Finsler geometry of classical Lie groups (1997)
For a (compact) subset $K$ of a metric space and $\varepsilon > 0$, the {\em covering number} $N(K , \varepsilon )$ is defined as the smallest number of balls of radius $\varepsilon$ whose union...
Szarek, Stanislaw J., Werner, Elisabeth
Let $\mu$ be a Gaussian measure (say, on ${\bf R}^n$) and let $K, L \subset {\bf R}^n$ be such that K is convex, $L$ is a "layer" (i.e. $L = \{x : a \leq < x,u > \leq b \}$ for some $a$, $b \in {\bf...
Stanislaw Szarek And, Stanislaw J. Szarek, Elisabeth Werner
. Let ¯ be a Gaussian measure (say, on R n ) and let K;L ` R n be such that K is convex, L is a "layer" (i.e. L = fx : a hx; ui bg for some a, b 2 R and u 2 R n ) and the centers of mass...
Metric Entropy of Homogeneous Spaces (1997)
. For a (compact) subset K of a metric space and " ? 0, the covering number N(K; ") is defined as the smallest number of balls of radius " whose union covers K. Knowledge of the metric...
Volumes of Restricted Minkowski Sums and the Free Analogue of the Entropy Power Inequality (1995)
Szarek, Stanislaw J., Voiculescu, D.
In noncommutative probability theory independence can be based on free products instead of tensor products. This yields a highly noncommutative theory: free probability . Here we show that the...
Lattice coverings and gaussian measures of n-dimensional convex bodies (1995)
Banaszczyk, W., Szarek, Stanislaw J.
Let $\| \cdot \|$ be the euclidean norm on ${\bf R}^n$ and $\gamma_n$ the (standard) Gaussian measure on ${\bf R}^n$ with density $(2 \pi )^{-n/2} e^{- \| x\|^2 /2}$. Let $\vartheta$ ($ \simeq...
Random Banach spaces. The limitations of the method (1993)
Mankiewicz, P., Szarek, Stanislaw J.
We study the properties of "generic", in the sense of the Haar measure on the corresponding Grassmann manifold, subspaces of l^N_infinity of given dimension. We prove that every "well bounded"...
A Nonsymmetric Correlation Inequality for Gaussian Measure
Szarek, Stanislaw J., Werner, Elisabeth
Let[mu]be a Gaussian measure (say, onRn) and letK,L[subset, double equals]Rnbe such thatKis convex,Lis a "layer" (i.e.,L={x:Â a[less-than-or-equals, slant][less-than-or-equals, slant]b} for...