arXiv subjects
Publications and source records attributed to Eric L. Grinberg.
Motivations and derivations for Heron's triangle area formula abound. But how would one \emph{discover} the formula? We recount the challenge of designing a laboratory experiment to discover Heron's formula, and of doing so in the morning, in time for a noon class---a snapshot of collaboration with AI in fall 2025. We then reflect on extensions and generalizations.
We propose an interpretation of, and approach to, Helly's theorem that can be included quite early in the undergraduate curriculum. At the same time, the approach connects with contemporary models of data privacy and with sampling methods used in epidemiology. The presentation is intended to be accessible to teachers and their students.
We present a proof the Steiner-Lehmus equal bisectors theorem by applying the Law of sines in rapid succession to a side-by-side comparison. For nearly two centuries, the quest for a direct proof has sustained interest in proving and reproving this theorem. We suggest that a second driving force may also be at play.
Using a left-to-right "sweeping" algorithm, we define the \emph{Gauche basis} for the column space of a matrix $M$. By means of the Gauche basis we interpret the row reduced echelon form of $M$, and give a direct proof of its uniqueness. We conclude with pedagogical reflections.
Reopening a cold case, inspector Echelon, high-ranking in the Row Operations Center, is searching for a lost linear map, known to be nilpotent. When a partially decomposed matrix is unearthed, he reconstructs its reduced form, finding it singular. But were its roots nilpotent?
We invoke the Law of Sines to prove Morley's Trisector Theorem. Though the sinusoidal function appears, the proof is safe for the trigonometrically distanced.
We consider the Radon transform along lines in an $n$ dimensional vector space over the two element field. It is well known that this transform is injective and highly overdetermined. We classify the minimal collections of lines for which the restricted Radon transform is also injective. This is an instance of I.M.~Gelfand's {\it admissibility problem}. The solution is in stark contrast to the more uniform cases of the affine hyperplane transform and the projective line transform, which are addressed in other papers, \cite{Feld-G,Gr1}. The presentation here is intended to be widely accessible, requiring minimum background.
We consider a finite field model of the X-ray transform that integrates functions along lines in dimension 3, within the context of finite fields. The admissibility problem asks for minimal sets of lines for which the restricted transform is invertible. Graph theoretic conditions are known which characterize admissible collections of lines, and these have been counted using a brute force computer program. Here we perform the count by hand and, at the same time, produce a detailed illustration of the possible structures of inadmissible complexes. The resulting scrapbook may be of interest in an artificial intelligence approach to enumerating and illustrating admissible complexes in arbitrary dimensions (arbitrarily large ambient spaces, with transforms integrating over subspaces of arbitrary dimensions.)
We consider the problem of comparing the volumes of two star bodies in an even-dimensional euclidean space $\mathbb R^{2n} = \mathbb C^n$ by comparing their cross sectional areas along complex lines (special 2-dimensional real planes) through the origin. Under mild symmetry conditions on one of the bodies a Busemann-Petty type theorem holds. Quaternionic and Octonionic analogs also hold. The argument relies on integration in polar coordinates coupled with Jensen's inequality. Along the way we provide a criterion that detects which centered bodies are {\it circular}. i.e., stabilized by multiplication by complex numbers of unit modulus. Our goal is to present a Busemann-Petty type result with a minimum of required background and, in addition, to suggest characterizations of classes of star bodies by means of integral geometric inequalities.
We consider the X-ray transform in a projective space over a finite field. It is well known (after E. Bolker) that this transform is injective. We formulate an analog of I.M. Gelfand's admissibility problem for the Radon transform, which asks for a classification of all minimal sets of lines for which the restricted Radon transform is injective. The solution involves doubly ruled quadric surfaces.
Let $M$ be a Riemannian globally symmetric space of compact type, $M'$ its set of maximal flat totally geodesic tori, and $\mathrm{ad}(M)$ its adjoint space. We show that the kernel of the maximal flat Radon transform $τ:L^2(M) \rightarrow L^2(M')$ is precisely the orthogonal complement of the image of the pullback map $L^2(\mathrm{ad}(M))\rightarrow L^2(M)$. In particular, we show that the maximal flat Radon transform is injective if and only if $M$ coincides with its adjoint space.
In 1963, K.P.Grotemeyer proved an interesting variant of the Gauss-Bonnet Theorem. Let M be an oriented closed surface in the Euclidean space R^3 with Euler characteristic χ(M), Gauss curvature G and unit normal vector field n. Grotemeyer's identity replaces the Gauss-Bonnet integrand G by the normal moment ^2G, where $a$ is a fixed unit vector. Grotemeyer showed that the total integral of this integrand is (2/3)pi times chi(M). We generalize Grotemeyer's result to oriented closed even-dimesional hypersurfaces of dimension n in an (n+1) ndimensional space form N^{n+1}(k).