The covariogram problem
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Gabriele Bianchi
Abstract
The covariogram gX of a measurable set X in ℝn is the function which associates to each x Є ℝn the measure of the intersection of X with X + x. We are interested in understanding what information about a set can be obtained from its covariogram. Matheron asked whether a convex body K is determined from the knowledge of gK , and this is known as the covariogram problem. The covariogram appears in very different contexts, and the covariogram problem can be rephrased in different terms. For instance, it is equivalent to determining the characteristic function 1K of K from the modulus of its Fourier transform ̂1K in ℝn, a particular instance of the phase retrieval problem. The covariogram problem has also a discrete counterpart. We survey the known results and the methods.
Abstract
The covariogram gX of a measurable set X in ℝn is the function which associates to each x Є ℝn the measure of the intersection of X with X + x. We are interested in understanding what information about a set can be obtained from its covariogram. Matheron asked whether a convex body K is determined from the knowledge of gK , and this is known as the covariogram problem. The covariogram appears in very different contexts, and the covariogram problem can be rephrased in different terms. For instance, it is equivalent to determining the characteristic function 1K of K from the modulus of its Fourier transform ̂1K in ℝn, a particular instance of the phase retrieval problem. The covariogram problem has also a discrete counterpart. We survey the known results and the methods.
Kapitel in diesem Buch
- Frontmatter I
- Contents V
- Algebraically integrable bodies and related properties of the Radon transform 1
- The covariogram problem 37
- The logarithmic Minkowski conjecture and the Lp-Minkowski problem 83
- Bellman functions and continuous time 119
- Volume product 163
- Inequalities for sections and projections of convex bodies 223
- Borderline estimates for weighted singular operators and concavity 257
- Extremal sections and projections of certain convex bodies: a survey 343
- When does e−/τ/ maximize Fourier extension for a conic section? 391
- Affine surface area 427
- Analysis and geometry near the unit ball: proofs, counterexamples, and open questions 445
- Index 469
Kapitel in diesem Buch
- Frontmatter I
- Contents V
- Algebraically integrable bodies and related properties of the Radon transform 1
- The covariogram problem 37
- The logarithmic Minkowski conjecture and the Lp-Minkowski problem 83
- Bellman functions and continuous time 119
- Volume product 163
- Inequalities for sections and projections of convex bodies 223
- Borderline estimates for weighted singular operators and concavity 257
- Extremal sections and projections of certain convex bodies: a survey 343
- When does e−/τ/ maximize Fourier extension for a conic section? 391
- Affine surface area 427
- Analysis and geometry near the unit ball: proofs, counterexamples, and open questions 445
- Index 469
