arXiv subjects
Publications and source records attributed to Mariia Dospolova.
Using a higher-dimensional analog of an identity known to Kronecker, we discover a new Andrews--Crandall-type identity and use it to count the number of integer solutions to $x^2+2y^2+2z^2=n$.
Let $K$ be a convex compact $GB$-subset of a separable Hilbert space $H$. Denote by $\mathrm{Spec}_k K$ the set $\{(ξ_1(h), \ldots, ξ_k(h))\colon h\in K\}\subset \mathbb{R}^k,$ where $ξ_1, \ldots, ξ_k$ are independent copies of the isonormal Gaussian process on $H$. Tsirelson showed that in this case the intrinsic volumes of $K$ satisfy the relation \begin{equation*} V_k(K)= \frac{(2π)^{k/2}}{k!κ_k} \mathbf{E}\,\mathrm{Vol}_k(\mathrm{Spec}_k K). \end{equation*} Here, $\mathbf{E} \ \mathrm{Vol}_k(\mathrm{Spec}_k K)$ is the mean volume of $\mathrm{Spec}_k K$ and $κ_k$ is the volume of the $k$-dimensional unit ball. In this work, we generalize Tsirelson's theorem to the mixed volumes of the infinite-dimensional convex compact $GB$-subsets of $H$, first introducing this notion. Moreover, using the obtained result we compute the mixed volume of the closed convex hulls of the two orthogonal Wiener spirals.
For a convex lattice polytope $P\subset \mathbb R^d$ of dimension $d$ with vertices in $\mathbb Z^d$, denote by $L(P)$ its discrete volume which is defined as the number of integer points inside $P$. The classical result due to Ehrhart says that for a positive integer $n$, the function $L(nP)$ is a polynomial in $n$ of degree $d$ whose leading coefficient is the volume of $P$. In particular, $L(nP)$ approximates the volume of $nP$ for large $n$. In convex geometry, one of the central notion which generalizes the volume is the intrinsic volumes. The main goal of this paper is to introduce their discrete counterparts. In particular, we show that for them the analogue of the Ehrhart result holds, where the volume is replaced by the intrinsic volume. We also introduce and study a notion of Grassmann valuation which generalizes both the discrete volume and the solid-angle valuation introduced by Reeve and Macdonald.