arXiv subjects

Alexander Koldobsky

Publications and source records attributed to Alexander Koldobsky.

49 records · Page 3Linked to original sources

arXiv1999

An analytic solution to the Busemann-Petty problem on sections of convex bodies

We derive a formula connecting the derivatives of parallel section functions of an origin-symmetric star body in R^n with the Fourier transform of powers of the radial function of the body. A parallel section function (or (n-1)-dimensional X-ray) gives the ((n-1)-dimensional) volumes of all hyperplane sections of the body orthogonal to a given direction. This formula provides a new characterization of intersection bodies in R^n and leads to a unified analytic solution to the Busemann-Petty problem: Suppose that K and L are two origin-symmetric convex bodies in R^n such that the ((n-1)-dimensional) volume of each central hyperplane section of K is smaller than the volume of the corresponding section of L; is the (n-dimensional) volume of K smaller than the volume of L? In conjunction with earlier established connections between the Busemann-Petty problem, intersection bodies, and positive definite distributions, our formula shows that the answer to the problem depends on the behavior of the (n-2)-nd derivative of the parallel section functions. The affirmative answer to the Busemann-Petty problem for n\le 4 and the negative answer for n\ge 5 now follow from the fact that convexity controls the second derivatives, but does not control the derivatives of higher orders.

math.MG↗

arXiv1997

Second derivative test for isometric embeddings in $L_p$

An old problem of P. Levy is to characterize those Banach spaces which embed isometrically in $L_p.$ We show a new criterion in terms of the second derivative of the norm. As an application, we show that if $M$ is a twice differentiable Orlicz function with $M'(0)=M''(0)=0$ then the $n$-dimensional Orlicz space $\ell_M^n,\ n\ge 3,$ does not embed isometrically in $L_p$ with $0<p\le 2.$ These results generalize and clear up the recent solution to the 1938 Schoenberg's problem on positive definite functions.

math.FA↗

arXiv1996

Positive definite distributions and subspaces of $L_{-p}$ with applications to stable processes

We define embedding of an $n$-dimensional normed space into $L_{-p},\ 0<p<n$ by extending analytically with respect to $p$ the corresponding property of the classical $L_p$-spaces. The well-known connection between embeddings into $L_p$ and positive definite functions is extended to the case of negative $p$ by showing that a normed space embeds in $L_{-p}$ if and only if $\|x\|^{-p}$ is a positive definite distribution. Using this criterion, we generalize the recent solutions to the 1938 Schoenberg's problems by proving that the spaces $\ell_q^n,\ 2<q\le \infty$ embed in $L_{-p}$ if and only if $p\in [n-3,n).$ We show that the technique of embedding in $L_{-p}$ can be applied to stable processes in some situations where standard methods do not work. As an example, we prove inequalities of correlation type for the expectations of norms of stable vectors. In particular, for every $p\in [n-3,n),$ $\Bbb E(\max_{i=1,...,n} |X_i|^{-p}) \ge \Bbb E(\max_{i=1,...,n} |Y_i|^{-p}),$ where $X_1,...,X_n$ and $Y_1,...,Y_n$ are jointly $q$-stable symmetric random variables, $0<q\le 2,$ so that, for some $k\in \Bbb N,\ 1\le k <n,$ the vectors $(X_1,...,X_k)$ and $(X_{k+1},...,X_n)$ have the same distributions as $(Y_1,...,Y_k)$ and $(Y_{k+1},...,Y_n),$ respectively, but $Y_i$ and $Y_j$ are independent for every choice of $1\le i\le k,\ k+1\le j\le n.$

math.FA↗

arXiv1996

Inverse formula for the Blaschke-Levy representation with applications to zonoids and sections of star bodies

We say that an even continuous function $H$ on the unit sphere $Ω$ in $R^n$ admits the Blaschke-Levy representation with $q>0$ if there exists an even function $b\in L_1(Ω)$ so that $H^q(x)=\int_Ω|(x,ξ)|^q b(ξ)\ dξ$ for every $x\in Ω.$ This representation has numerous applications in convex geometry, probability and Banach space theory. In this paper, we present a simple formula (in terms of the derivatives of $H$) for calculating $b$ out of $H.$ We use this formula to give a sufficient condition for isometric embedding of a space into $L_p$ which contributes to the 1937 P.Levy's problem and to the study of zonoids. Another application gives a Fourier transform formula for the volume of $(n-1)$-dimensional central sections of star bodies in $R^n.$ We apply this formula to find the minimal and maximal volume of central sections of the unit balls of the spaces $\ell_p^n$ with $0<p<2.$

math.FA↗

arXiv1996

A correlation inequality for the expectations of norms of stable vectors

For $0<q\le 2,\ 1\le k < n,$ let $X=(X_1,...,X_n)$ and $Y=(Y_1,...,Y_n)$ be symmetric $q$-stable random vectors so that the joint distributions of $X_1,...,X_k$ and $X_{k+1},...,X_n$ are equal to the joint distributions of $Y_1,...,Y_k$ and $Y_{k+1},...,Y_n,$ respectively, but $Y_i$ and $Y_j$ are independent for every $1\le i \le k,\ k+1\le j \le n.$ We prove that $\Bbb E (f(X)) \ge \Bbb E (f(Y))$ where $f$ is any continuous, positive, homogeneous of the order $p\in (-n,0)$ function on $\Bbb R^n\setminus \{0\}$ such that $f$ is a positive definite distribution in $\Bbb R^n,$ and $f(u,v)=f(u,-v)$ for every $u\in \Bbb R^k,\ v\in \Bbb R^{n-k}.$ As a particular case, we show that $$\Bbb E\ (\max_{i=1,...,n} |X_i|)^p \ge \Bbb E\ (\max_{i=1,...,n} |Y_i|)^p$$ for every $p\in (-n,-n+1).$ The latter inequality is related to Slepian's Lemma and to the Gaussian correlation problem.

math.PR↗

arXiv1995

Inequalities of correlation type for symmetric stable random vectors

We prove that, for any jointly stable random variables $X_1, \dots, X_k$ with zero mean, any $m<k,$ and any even continuous positive definite functions $f$ and $g$ on $\Bbb R^m$ and $\Bbb R^{k-m},$ the random variables $f(X_1,\dots,X_m)$ and $g(X_{m+1}, \dots,X_k)$ are non-negatively correlated. We also show another result that is related to an old question of whether $$P(\max_{1\le i\le k} |X_i|<t) \ge P(\max_{1\le i\le m} |X_i|<t) \ P(\max_{m+1\le i\le k} |X_i|<t)$$ where $X_1,\dots,X_k$ are jointly Gaussian random variables with zero mean, and $m<k.$ We show that the quantity in the left-hand side has a local minimum at the point where the random variables $X_i$ and $X_j$ are independent for any choice of $1\le i\le m$ and $m+1\le j\le k.$

math.FA↗

arXiv1994

A Banach subspace of $L_{1/2}$ which does not embed in $L_1$ (isometric version)

For every $n\geq 3,$ we construct an $n$-dimensional Banach space which is isometric to a subspace of $L_{1/2}$ but is not isometric to a subspace of $L_1.$ The isomorphic version of this problem (posed by S. Kwapien in 1969) is still open. Another example gives a Banach subspace of $L_{1/4}$ which does not embed isometrically in $L_{1/2}.$ Note that, from the isomorphic point of view, all the spaces $L_q$ with $q<1$ have the same Banach subspaces.

math.FA↗

arXiv1993

Isometries of $L_p$-spaces of solutions of homogeneous partial differential equations

Let $ n\geq 2, A=(a_{ij})_{i,j=1}^{n}$ be a real symmetric matrix, $a=(a_i)_{i=1}^{n}\in \Bbb R^n.$ Consider the differential operator $D_A = \sum_{i,j=1}^n a_{ij}{\partial^2 \over \partial x_i \partial x_j}+ \sum_{i=1}^n a_i{\partial \over \partial x_i}.$ Let $E$ be a bounded domain in $\Bbb R^n,$ $p>0.$ Denote by $L_{D_A}^p(E)$ the space of solutions of the equation $D_A f=0$ in the domain $E$ provided with the $L_p$-norm. We prove that, for matrices $A,B,$ vectors $a,b,$ bounded domains $E,F,$ and every $p>0$ which is not an even integer, the space $L_{D_A}^p(E)$ is isometric to a subspace of $L_{D_B}^p(F)$ if and only if the matrices $A$ and $B$ have equal signatures, and the domains $E$ and $F$ coincide up to a natural mapping which in the most cases is affine. We use the extension method for $L_p$-isometries which reduces the problem to the question of which weighted composition operators carry solutions of the equation $D_A f=0$ in $E$ to solutions of the equation $D_B f=0$ in $F.$

math.FA↗

arXiv1993

The Fourier transform of order statistics with applications to Lorentz spaces

We present a formula for the Fourier transforms of order statistics in $\Bbb R^n$ showing that all these Fourier transforms are equal up to a constant multiple outside the coordinate planes in $\Bbb R^n.$ For $a_1\geq ... \geq a_n\ge0$ and $q>0,$ denote by $\ell_{w,q}^n$ the $n$-dimensional Lorentz space with the norm $\|(x_1,...,x_n)\| = (a_1 (x_1^{*})^q +...+ a_n (x_n^{*})^q)^{1/q}$, where $(x_1^{*},...,x_n^{*})$ is the non-increasing permutation of the numbers $|x_1|,...,|x_n|.$ We use the above mentioned formula and the Fourier transform criterion of isometric embeddability of Banach spaces into $L_q$ \cite{10} to prove that, for $n\geq 3$ and $q\leq 1,$ the space $\ell_{w,q}^n$ is isometric to a subspace of $L_q$ if and only if the numbers $a_1,...,a_n$ form an arithmetic progression. For $q>1,$ all the numbers $a_i$ must be equal so that $\ell_{w,q}^n = \ell_q^n.$ Consequently, the Lorentz function space $L_{w,q}(0,1)$ is isometric to a subspace of $L_q$ if and only if {\it either} $0<q<\infty$ and the weight $w$ is a constant function (so that $L_{w,q}= L_q$), {\it or} $q\le 1$ and $w(t)$ is a decreasing linear function. Finally, we relate our results to the theory of positive definite functions.

math.FA↗

arXiv1993

Isometric stability property of certain Banach spaces

Let $E$ be one of the spaces $C(K)$ and $L_1$, $F$ be an arbitrary Banach space, $p>1,$ and $(X,σ)$ be a space with a finite measure. We prove that $E$ is isometric to a subspace of the Lebesgue-Bochner space $L_p(X;F)$ only if $E$ is isometric to a subspace of $F.$ Moreover, every isometry $T$ from $E$ into $L_p(X;F)$ has the form $Te(x)=h(x)U(x)e, e\in E,$ where $h:X\rightarrow R$ is a measurable function and, for every $x\in X,$ $U(x)$ is an isometry from $E$ to $F.$

math.FA↗

arXiv1992

Operators preserving orthogonality are isometries

Let $E$ be a real Banach space. For $x,y \in E,$ we follow R.James in saying that $x$ is orthogonal to $y$ if $\|x+αy\|\geq \|x\|$ for every $α\in R$. We prove that every operator from $E$ into itself preserving orthogonality is an isometry multiplied by a constant.

math.FA↗

arXiv1992

Common subspaces of $L_{p}$-spaces

For $n\geq 2, p<2$ and $q>2,$ does there exist an $n$-dimensional Banach space different from Hilbert spaces which is isometric to subspaces of both $L_{p}$ and $L_{q}$? Generalizing the construction from the paper "Zonoids whose polars are zonoids" by R.Schneider we give examples of such spaces. Moreover, for any compact subset $Q$ of $(0,\infty)\setminus \{2k, k\in N\},$ we can construct a space isometric to subspaces of $L_{q}$ for all $q\in Q$ simultaneously. This paper requires vanilla.sty

math.FA↗

arXiv1992

Schoenberg's Problem on Positive Definite Functions

If $n \ge 3$, $q>2$ and $β> 0$ then the function $\exp(-(|x_1|^q+|x_2|^q+\dots+|x_n|^q)^{β/q})$\ is not positive definite. This result gives an answer to a question posed by I.J.~Schoenberg in 1938. This text is an authorized English translation of the paper published in Russian in Algebra and Analysis 3(1991), \#3, p.78--85.

math.FA↗