Intrinsic volumes of Sobolev balls with applications to Brownian convex hulls (Q2822723)

The authors study intrinsic volumes of infinite-dimensional convex compact sets both in deterministic and in random settings. The aim of this work is to provide explicit formulas for intrinsic volumes of some class of sets. Those sets are unit balls with respect to Sobolev-type seminorms, ellipsoids in the Hilbert space and also Brownian convex hulls, Brownian zonoids.NEWLINENEWLINERecall that intrinsic volumes of finite-dimensional set \(T\subset\mathbb R^n\) can be defined as the coefficients in the Steiner formula \(\operatorname{Vol}_n(T+rB_n) = \sum_{k=0}^{n} \kappa_{n-k}V_k(T)r^{n-k}\) whereas the Kubota formula allows calculating intrinsic volumes as a mean of volumes \(W_k\) of uniformly distributed \(k\)-dimensional projections of \(T\): \(V_k(T) = \binom{n}{k}\frac{\kappa_n}{\kappa_k\kappa_{n-k}}\operatorname{E} W_k\). Then, in the late seventies intrinsic volumes of infinite-dimensional were introduced: [\textit{V. N. Sudakov}, Proc. Steklov Inst. Math. 141, 178 p. (1979); translation from Tr. Mat. Inst. Steklov 141, 191 p. (1976; Zbl 0409.60005)] and [\textit{S. Chevet}, Z. Wahrscheinlichkeitstheor. Verw. Geb. 36, 47--65 (1976; Zbl 0326.60047)]. In particular, Sudakov's formula for the first intrinsic volume \(V_1(K) = \sqrt{2\pi}\operatorname{E}\sup X_t\) connects the mean weight of an infinite-dimensional convex compact set \(K\) with the maximum of the isonormal Gaussian process over this set. In this paper, the authors exploit this formula to derive explicit expressions for intrinsic volumes of Sobolev balls. Thus, intrinsic volumes of unit Lipschitz balls are calculated \(V_k = \frac{\pi^{k/2}}{\Gamma(\frac{3}{2}k+1)}\) (case \(p=\infty\)), and the first intrinsic volume of a Sobolev ball is \(V_1(\mathbb K^p) = \sqrt{2\pi}\operatorname{E}\|X\|_q\), where \(X\) is a particular Gaussian process and \(\frac{1}{p}+\frac{1}{q}=1\) (case \(p\in[1,\infty]\)). The work contains a wide range of examples of that kind.NEWLINENEWLINEThe authors use results of [\textit{F. Gao} and \textit{R. A. Vitale}, Discrete Comput. Geom. 26, No. 1, 41--50 (2001; Zbl 0986.60006)] and [\textit{F. Gao}, Electron. Commun. Probab. 8, Paper No. 1, 1--5 (2003; Zbl 1125.60303)] to give a new proof for the formula of expected volume of Brownian convex hull due to [\textit{R. Eldan}, Electron. J. Probab. 19, Paper No. 45, 34 p. (2014; Zbl 1298.52005)] \(\operatorname{E}(\operatorname {Vol}_n(K)) = (\frac{\pi}{2})^{\frac{n}{2}}\frac{1}{\Gamma(\frac{n}{2}+1)^2}\).