A note on subgaussian estimates for linear functionals on convex bodies
نویسندگان
چکیده
We give an alternative proof of a recent result of Klartag on the existence of almost subgaussian linear functionals on convex bodies. If K is a convex body in Rn with volume one and center of mass at the origin, there exists x 6= 0 such that |{y ∈ K : |〈y, x〉| > t‖〈·, x〉‖1}| 6 exp(−ct 2/ log2(t+ 1)) for all t > 1, where c > 0 is an absolute constant. The proof is based on the study of the Lq–centroid bodies of K. Analogous results hold true for general log-concave measures.
منابع مشابه
A class of certain properties of approximately n-multiplicative maps between locally multiplicatively convex algebras
We extend the notion of approximately multiplicative to approximately n-multiplicative maps between locally multiplicatively convex algebras and study some properties of these maps. We prove that every approximately n-multiplicative linear functional on a functionally continuous locally multiplicatively convex algebra is continuous. We also study the relationship between approximately mu...
متن کاملA New Affine Invariant for Polytopes and Schneider’s Projection Problem
New affine invariant functionals for convex polytopes are introduced. Some sharp affine isoperimetric inequalities are established for the new functionals. These new inequalities lead to fairly strong volume estimates for projection bodies. Two of the new affine isoperimetric inequalities are extensions of Ball’s reverse isoperimetric inequalities. If K is a convex body (i.e., a compact, convex...
متن کاملLearning sub-Gaussian classes : Upper and minimax bounds
Most the results contained in this note have been presented at the SMF meeting, which took place in May 2011; the rest have been obtained shortly after the time of the meeting. The question we study has to do with the optimality of Empirical Risk Minimization as a learning procedure in a convex class – when the problem is subgaussian. Subgaussian learning problems are a natural object because t...
متن کاملTowards a Constructive Version of Banaszczyk's Vector Balancing Theorem
An important theorem of Banaszczyk (Random Structures & Algorithms ‘98) states that for any sequence of vectors of `2 norm at most 1/5 and any convex body K of Gaussian measure 1/2 in R, there exists a signed combination of these vectors which lands inside K. A major open problem is to devise a constructive version of Banaszczyk’s vector balancing theorem, i.e. to find an efficient algorithm wh...
متن کاملThe Shattering dimension of sets of linear functionals
We evaluate the shattering dimension of various classes of linear functionals on various symmetric convex sets. This is applied in two different directions. The first is the determination of whether or not several classes of linear functionals satisfy the uniform Central Limit Theorem or the uniform Law of Large Numbers. The second direction, which was the main motivation of this study, is the ...
متن کامل