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Published online by Cambridge University Press: 27 June 2025
We introduce the mixed convolution bodies of two convex symmetric bodies. We prove that if the boundary of a body K is smooth enough then as δ tends to 1 the δ-M* -convolution body of K with itself tends to a multiple of the Euclidean ball after proper normalization. On the other hand we show that the δ-M*-convolution body of the n-dimensional cube is homothetic to the unit ball of 𝓁n1.
1. Introduction
Throughout this note K and L denote convex symmetric bodies in R.n. Our notation will be the standard notation that can be found, for example, in [2] and [4]. For 1 ≤ m ≤ n, Vm(K) denotes the m-th mixed volume of K (i.e., mixing m copies of K with n - m copies of the Euclidean ball Bn of radius one in R.n). Thus if m = n then Vn(K) = voln(K) and if m = 1 then V1(K) = ω(K) the mean width of K. For 0 < δ < 1 we define the m-th mixed 8-convolution body of the convex symmetric bodies K and L in R.n: DEFINITION. The m-th mixed δ-convolution body of K and L is defined to be the set It is a consequence of the Brunn-Minkowski inequality for mixed volumes that these bodies are convex.
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