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We prove that the unit cube B ∞ is a strict local minimizer for the Mahler volume product voln(K)voln(K) in the class of origin symmetric convex bodies endowed with the Banach-Mazur distance.
The intersection body of a ball is again a ball. So, the unit ball Bd ⊂ Rd is a fixed point of the intersection body operator acting on the space of all star-shaped origin symmetric bodies endowed… (More)
In [F] Firey extended the notion of the Minkowski sum, and introduced, for each real p, a new linear combination of convex bodies, that he called p-sums. Lutwak [Lu2], [Lu3] showed that these Firey… (More)
Abstract. The Fourier analytic approach to sections of convex bodies has recently been developed and has led to several results, including a complete analytic solution to the BusemannPetty problem,… (More)
Let f and g be two continuous functions on the unit sphere Sn−1 in Rn , n ≥ 3, and let their restrictions to any one-dimensional great circle E coincide after some rotation φE of this circle: f (φE… (More)
It has been noticed long ago that many results on sections and projections are dual to each other, though methods used in the proofs are quite different and don't use the duality of underlying… (More)
In this note we reconstruct a convex body of revolution from the areas of its shadows by giving a precise formula for the support function.
It is well-known (see, for example, , , and references therein) that f ∈ H(D), p > 0, if Mp(r, f) ≤ C < ∞. It is also well-known that f ∈ H(D) if and only if a subharmonic function |f(z)| has… (More)
For 0 ≤ α < 1 we construct examples of even integrable functions Ω on the unit sphere S with mean value zero satisfying
We discuss some open questions on unique determination of convex bodies.