Recently, motivated by the rapid increase of the data size in various applications, Monemizadeh [APPROX'23] and Driemel, Monemizadeh, Oh, Staals, and Woodruff [SoCG'25] studied geometric problems in the setting where the only access to the input point set is via querying a range-search oracle. Algorithms in this setting are evaluated on two criteria: (i) the number of queries to the oracle and (ii) the error of the output. In this paper, we continue this line of research and investigate one of the most fundamental geometric problems in the oracle setting, i.e., the convex hull problem. Let P be an unknown set of points in [0,1]^d equipped with a range-emptiness oracle. Via querying the oracle, the algorithm is supposed to output a convex polygon C ⊆ [0,1]^d as an estimation of the convex hull CH(P) of P. The error of the output is defined as the volume of the symmetric difference C ⊕ CH(P) = (C∖CH(P)) ∪ (CH(P)∖C). We prove tight and near-tight tradeoffs between the number of queries and the error of the output for different variants of the problem, depending on the type of the range-emptiness queries and whether the queries are non-adaptive or adaptive. - Orthogonal emptiness queries in d-dimensional space: We show that the minimum error a deterministic algorithm can achieve with q queries is Θ(q^{-1/d}) if the queries are non-adaptive, and Θ(q^{-1/(d-1)}) if the queries are adaptive. In particular, in 2D, the bounds are Θ(1/√q) and Θ(1/q) for non-adaptive and adaptive queries, respectively. - Halfplane emptiness queries in 2D: We show that the minimum error a deterministic algorithm can achieve with q queries is Θ(1/√q) if the queries are non-adaptive, and Θ̃(1/q²) if the queries are adaptive. Here Θ̃(⋅) hides logarithmic factors.

We study two fundamental geometric problems on a dynamic set of n axis-parallel boxes in d-dimensional space. The maximum depth problem asks for the largest number of boxes that contain a common point, whereas Klee’s measure problem asks for the volume of the union of the boxes. We present fully dynamic exact data structures for both problems achieving Õ(n^{(d-1)/2}) amortized update time. This update time is optimal for an exact dynamic algorithm, up to logarithmic factors, assuming the Combinatorial k-Clique Hypothesis. Previously, matching bounds were established only for d = 1 [Imai and Asano, J. Algo.'83], and for d = 2 [Suri, Xue, Yang, and Zhu, SoCG'25]. Our approach integrates a classic grid-based partition framework with a novel charging analysis that controls the cost of structure-sensitive offline routines within each cell. This argument allows us to perform a global aggregation of the update time, by circumventing the worst-case costs associated with individual cell updates. We believe this technique may be of independent interest for other dynamic geometric problems.