On combinatorial compexity of convex sequences

Recent Results on Douglas-Rachford Methods for Combinatorial Optimization Problems

We discuss recent positive experiences applying convex feasibility algorithms of Douglas--Rachford type to highly combinatorial and far from convex problems.

非阿基米德凸集的组合性质

Combinatorial properties of non-archimedean convex sets

本文研究任意赋值域上凸集的组合性质，通过球完备赋值域中赋值环上模的组合描述，证明了分数 Helly 定理和 Bárány 定理的非阿基米德类比，并揭示了有限宽度和 VC 维数等实数域凸集不具备的新性质。

Calculus proofs of some combinatorial inequalities

Using calculus we show how to prove some combinatorial inequalities of the type log-concavity or log-convexity. It is shown by this method that binomial coefficients and Stirling numbers of the first and second kinds are log-concave, and that Motzkin numbers and secondary structure numbers of rank 1 are log-convex. In fact, we prove via calculus a much stronger result that a natural continuous ``patchwork'' (i.e. corresponding dynamical systems) of Motzkin numbers and secondary structures recursions are increasing functions. We indicate how to prove asymptotically the log-convexity for general…

Six combinatorial clases of maximal convex tropical polyhedra

In this paper we bring together tropical linear algebra and convex 3-dimensional bodies. We show how certain convex 3-dimensional bodies having 20 vertices and 12 facets can be encoded in a $4\times 4$ integer zero-diagonal matrix $A$. A tropical tetrahedron is the set of points in $\R^3$ tropically spanned by four given tropically non-coplanar points. It is a near-miss Johnson solid. The coordinates of the points are arranged as the columns of a $4\times 4$ real matrix $A$ and the tetrahedron is denoted $\spann(A)$. We study tropical tetrahedra which are convex and maximal, computing the extr…

Convex Hulls, Oracles, and Homology

This paper presents a new algorithm for the convex hull problem, which is based on a reduction to a combinatorial decision problem POLYTOPE-COMPLETENESS-COMBINATORIAL, which in turn can be solved by a simplicial homology computation. Like other convex hull algorithms, our algorithm is polynomial (in the size of input plus output) for simplicial or simple input. We show that the ``no''-case of POLYTOPE-COMPLETENESS-COMBINATORIAL has a certificate that can be checked in polynomial time (if integrity of the input is guaranteed).

On the Geometry of Holmsen's Combinatorial Version of the Colorful Carathéodory

Carathéodorys Theorem of convex hulls plays an important role in convex geometry. In 1982, Bárány formulated and proved a more general version, called the Colorful Carathéodory. This colorful version was even more generalized by Holmsen in 2016. He formulated a combinatorial extension and found a topological proof. Taking a dual point of view we gain an equivalent formulation of Holmsen's result that has a more geometric meaning.

Notes on Convex Sets, Polytopes, Polyhedra, Combinatorial Topology, Voronoi Diagrams and Delaunay Triangulations

Some basic mathematical tools such as convex sets, polytopes and combinatorial topology, are used quite heavily in applied fields such as geometric modeling, meshing, computer vision, medical imaging and robotics. This report may be viewed as a tutorial and a set of notes on convex sets, polytopes, polyhedra, combinatorial topology, Voronoi Diagrams and Delaunay Triangulations. It is intended for a broad audience of mathematically inclined readers. I have included a rather thorough treatment of the equivalence of V-polytopes and H-polytopes and also of the equivalence of V-polyhedra and H-poly…

Higher order log-monotonicity of combinatorial sequences

A sequence $\{z_n\}_{n\geq0}$ is called ratio log-convex in the sense that the ratio sequence $\{\frac{z_{n+1}}{z_n}\}_{n\geq0}$ is log-convex. Based on a three-term recurrence for sequences, we develop techniques for dealing with the ratio log-convexity of ratio sequences. As applications, we prove that the ratio sequences of numbers, including the derangement numbers, the Motzkin numbers, the Fine numbers, Franel numbers and the Domb numbers are ratio log-convex, respectively. Finally, we not only prove that the sequence of derangement numbers is asymptotically infinitely log-monotonic, but …

Quantitative combinatorial geometry for concave functions

We prove several exact quantitative versions of Helly's and Tverberg's theorems, which guarantee that a finite family of convex sets in $R^d$ has a large intersection. Our results characterize conditions that are sufficient for the intersection of a family of convex sets to contain a "witness set" which is large under some concave or log-concave measure. The possible witness sets include ellipsoids, zonotopes, and $H$-convex sets. Our results also bound the complexity of finding the best approximation of a family of convex sets by a single zonotope or by a single $H$-convex set. We obtain colo…

Prescribing Symmetries and Automorphisms for Polytopes

We study finite groups that occur as combinatorial automorphism groups or geometric symmetry groups of convex polytopes. When $Γ$ is a subgroup of the combinatorial automorphism group of a convex $d$-polytope, $d\geq 3$, then there exists a convex $d$-polytope related to the original polytope with combinatorial automorphism group exactly $Γ$. When $Γ$ is a subgroup of the geometric symmetry group of a convex $d$-polytope, $d\geq 3$, then there exists a convex $d$-polytope related to the original polytope with both geometric symmetry group and combinatorial automorphism group exactly $Γ$. These…

Characterizing circles by a convex combinatorial property

Let $K_0$ be a compact convex subset of the plane $\mathbb R^2$, and assume that $K_1\subseteq \mathbb R^2$ is similar to $K_0$, that is, $K_1$ is the image of $K_0$ with respect to a similarity transformation $\mathbb R^2\to\mathbb R^2$. Kira Adaricheva and Madina Bolat have recently proved that if $K_0$ is a disk and both $K_0$ and $K_1$ are included in a triangle with vertices $A_0$, $A_1$, and $A_2$, then there exist a $j\in \{0,1,2\}$ and a $k\in\{0,1\}$ such that $K_{1-k}$ is included in the convex hull of $K_k\cup(\{A_0,A_1, A_2\}\setminus\{A_j\})$. Here we prove that this property char…

Lattice convex chains in the plane

A detailed combinatorial analysis of planar lattice convex polygonal lines is presented. This makes it possible to answer an open question of Vershik regarding the existence of a limit shape when the number of vertices is constrained. The method which is used emphasizes the connection of the combinatorial analysis with the zeros of the zeta function. It is shown how the Riemann Hypothesis leads to an asymptotic equivalent of the number of convex chains.

Intrinsic Volumes of Polyhedral Cones: A combinatorial perspective

The theory of intrinsic volumes of convex cones has recently found striking applications in areas such as convex optimization and compressive sensing. This article provides a self-contained account of the combinatorial theory of intrinsic volumes for polyhedral cones. Direct derivations of the General Steiner formula, the conic analogues of the Brianchon-Gram-Euler and the Gauss-Bonnet relations, and the Principal Kinematic Formula are given. In addition, a connection between the characteristic polynomial of a hyperplane arrangement and the intrinsic volumes of the regions of the arrangement, …

Tuning as convex optimisation: a polynomial tuner for multi-parametric combinatorial samplers

Combinatorial samplers are algorithmic schemes devised for the approximate- and exact-size generation of large random combinatorial structures, such as context-free words, various tree-like data structures, maps, tilings, RNA molecules. They can be adapted to combinatorial specifications with additional parameters, allowing for a more flexible control over the output profile of parametrised combinatorial patterns. One can control, for instance, the number of leaves, profile of node degrees in trees or the number of certain sub-patterns in generated strings. However, such a flexible control req…

Betti numbers of the geometric spaces associated to nonrational simple convex polytopes

We compute the Betti numbers of the geometric spaces associated to nonrational simple convex polytopes and find that they depend on the combinatorial type of the polytope exactly as in the rational case. This shows that the combinatorial features of the starting polytope are encoded in these generalized toric spaces as they are in their rational counterparts.

偏序体与组合神经码

Oriented Matroids and Combinatorial Neural Codes

本文将偏序体理论与组合神经码理论联系起来，证明了判断组合码是否凸是NP难问题。

Convex hulls of spheres and convex hulls of convex polytopes lying on parallel hyperplanes

Given a set $Σ$ of spheres in $\mathbb{E}^d$, with $d\ge{}3$ and $d$ odd, having a fixed number of $m$ distinct radii $ρ_1,ρ_2,...,ρ_m$, we show that the worst-case combinatorial complexity of the convex hull $CH_d(Σ)$ of $Σ$ is $Θ(\sum_{1\le{}i\ne{}j\le{}m}n_in_j^{\lfloor\frac{d}{2}\rfloor})$, where $n_i$ is the number of spheres in $Σ$ with radius $ρ_i$. To prove the lower bound, we construct a set of $Θ(n_1+n_2)$ spheres in $\mathbb{E}^d$, with $d\ge{}3$ odd, where $n_i$ spheres have radius $ρ_i$, $i=1,2$, and $ρ_2\neρ_1$, such that their convex hull has combinatorial complexity $Ω(n_1n_2^{…

A Set of Questions in Combinatorial and Metric Geometry

We briefly introduce several problems: (1) a generalization of the convex fair partition conjecture, (2) on non-trivial invariants among polyhedrons that can be formed from the same set of face polygons, (3) two questions on assembling rectangular tiles to form larger rectangles and (4) on convex regions which maximize and minimize the diameter for specified area and perimeter. For each question, we discuss partial solutions and indicate aspects that to our knowledge, await exploration.

Tropical Convex Hull Computations

This is a survey on tropical polytopes from the combinatorial point of view and with a focus on algorithms. Tropical convexity is interesting because it relates a number of combinatorial concepts including ordinary convexity, monomial ideals, subdivisions of products of simplices, matroid theory, finite metric spaces, and the tropical Grassmannians. The relationship between these topics is explained via one running example throughout the whole paper. The final section explains how the new version 2.9.4 of the software system polymake can be used to compute with tropical polytopes.

Quantitative $(p,q)$ theorems in combinatorial geometry

We show quantitative versions of classic results in discrete geometry, where the size of a convex set is determined by some non-negative function. We give versions of this kind for the selection theorem of Bárány, the existence of weak epsilon-nets for convex sets and the $(p,q)$ theorem of Alon and Kleitman. These methods can be applied to functions such as the volume, surface area or number of points of a discrete set. We also give general quantitative versions of the colorful Helly theorem for continuous functions.

A note on the combinatorial structure of finite and locally finite simplicial complexes of nonpositive curvature

We investigate the collapsibility of systolic finite simplicial complexes of arbitrary dimension. The main tool we use in the proof is discrete Morse theory. We shall consider a convex subcomplex of the complex and project any simplex of the complex onto a ball around this convex subcomplex. These projections will induce a convenient gradient matching on the complex. Besides we analyze the combinatorial structure of both CAT(0) and systolic locally finite simplicial complexes of arbitrary dimensions. We will show that both such complexes possess an arborescent structure. Along the way we make …

Morphisms of Neural Codes

We define a notion of morphism between combinatorial codes, making the class of all combinatorial codes into a category $\mathbf{Code}$. We show that morphisms can be used to remove redundant information from a code, and that morphisms preserve convexity. This fact leads us to define "minimally non-convex" codes. We propose a program to characterize these minimal obstructions to convexity and hence characterize all convex codes. We implement a library of Sage code to perform computation with morphisms. These computational methods yield the smallest to-date example of a non-convex code with no …

Order-forcing in Neural Codes

Convex neural codes are subsets of the Boolean lattice that record the intersection patterns of convex sets in Euclidean space. Much work in recent years has focused on finding combinatorial criteria on codes that can be used to classify whether or not a code is convex. In this paper we introduce order-forcing, a combinatorial tool which recognizes when certain regions in a realization of a code must appear along a line segment between other regions. We use order-forcing to construct novel examples of non-convex codes, and to expand existing families of examples. We also construct a family of …

Order Types of convex bodies

We give new bounds on the Erdos-Szekeres theorems for convex bodies of Bisztriczky and Fejes Toth and of Pach and Toth. We derive them from a combinatorial characterization of convex position of a family of planar convex bodies. This characterization confirms that the concept of Order Type for points can be extended to noncrossing families of convex bodies in a geometrically meaningful way.

Strictly monotone sequences of lower and upper bounds on Perron values and their combinatorial applications

In this paper, we present monotone sequences of lower and upper bounds on the Perron value of a nonngeative matrix, and we study their strict monotonicity. Using those sequences, we provide two combinatorial applications. One is to improve bounds on Perron values of rooted trees in combinatorial settings, in order to find characteristic sets of trees. The other is to generate log-concave and log-convex sequences through the monotone sequences.

数字凸性与词上的组合学

Digital Convexity and Combinatorics on Words

本文从组合学角度研究数字凸词，形式化其几何性质，揭示其与 Christoffel 词及有限 Sturmian 词的联系，并分析膨胀与收缩运算。

Non-commutative groups as prescribed polytopal symmetries

We study properties of the realizations of groups as the combinatorial automorphism group of a convex polytope. We show that for any non-abelian group $G$ with a central involution there is a centrally symmetric polytope with $G$ as its combinatorial automorphisms. We show that for each integer $n$, there are groups that cannot be realized as the combinatorial automorphisms of convex polytopes of dimension at most $n$. We also give an optimal lower bound for the dimension of the realization of a group as the group of isometries that preserves a convex polytope.

凸可嵌入性与纽结理论

Convex Embeddability and Knot Theory

本文研究可数线性序的凸可嵌入性拟序及其诱导等价关系，获得组合与描述集合论结果，并扩展至循环序，进而应用于弧与纽结研究，确立几何对象间自然关系的组合性质及Borel归约复杂度下界。

A q-queens problem. VII. Combinatorial types of nonattacking chess riders

On a convex polygonal chessboard, the number of combinatorial types of nonattacking configuration of three identical chess riders with $r$ moves, such as queens, bishops, or nightriders, equals $r(r^2+3r-1)/3$, as conjectured by Chaiken, Hanusa, and Zaslavsky (2019). Similarly, for any number of identical 3-move riders the number of combinatorial types is independent of the actual moves.

Toric cohomological rigidity of simple convex polytopes

A simple convex polytope $P$ is \emph{cohomologically rigid} if its combinatorial structure is determined by the cohomology ring of a quasitoric manifold over $P$. Not every $P$ has this property, but some important polytopes such as simplices or cubes are known to be cohomologically rigid. In this article we investigate the cohomological rigidity of polytopes and establish it for several new classes of polytopes including products of simplices. Cohomological rigidity of $P$ is related to the \emph{bigraded Betti numbers} of its \emph{Stanley--Reisner ring}, another important invariants coming…

Combinatorial description of the principal congruence subgroup $γ$(2) in SL(2, z)

We characterize sequences of positive integers (c 1 , c 2 , ..., cn) for which the (2 x 2)-matrix c 1 --1 1 0 $\times$ $\times$ $\times$ cn --1 1 0 belongs to the principal congruence subgroup of level 2 in SL(2, Z). The answer is given in terms of dissections of a convex n-gon into a mixture of triangles and quadrilaterals.

Convexity of distinct sum sets

We study a combinatorial notion where given a set of lattice points one takes the set of all sums of subsets of a fixed size, and we ask if the given set comes from a convex lattice polytope whether the resulting set also comes from a convex lattice polytope. We obtain a positive result in dimension 2 and a negative result in higher dimensions. We apply this to the corner cut polyhedron.

A note on log-convexity of q-Catalan numbers

The q-Catalan numbers studied by Carlitz and Riordan are polynomials in q with nonnegative coefficients. They evaluate, at q=1, to the Catalan numbers: 1, 1, 2, 5, 14,..., a log-convex sequence. We use a combinatorial interpretation of these polynomials to prove a q-log-convexity result. The sequence of q-Catalan numbers is not q-log-convex in the narrow sense used by other authors, so our work suggests a more flexible definition of q-log convex be adopted.

Combinatorial Intersection Cohomology for Fans

We continue the approach toward a purely combinatorial "virtual" intersection cohomology for possibly non-rational fans, based on our investigation of equivariant intersection cohomology for toric varieties (see math.AG/9904159). Fundamental objects of study are "minimal extension sheaves" on "fan spaces". These are flabby sheaves of graded modules over a sheaf of polynomial rings, satisfying three relatively simple axioms that characterize the properties of the equivariant intersection cohomology sheaf on a toric variety, endowed with the finite topology given by open invariant subsets. These…

组合拼贴：从热带几何回归

Combinatorial patchworking: back from tropical geometry

本文将热带几何中的拼贴结果转回格点多面体三角剖分框架，证明其可推广至非凸情形，从而简化并推广实环簇 $T$-子流形拓扑的相关结论。

Combinatorial constructions of three-dimensional small covers

A small cover was introduced by Davis and Januszkiewicz as an $n$-dimensional closed manifold with a locally standard $Z_2)^n$-action such that its orbit space is a simple convex polytope. There exist a one-to-one correspondence between small covers and $(Z_2)^n$-colored polytopes. In this paper we study a construction of 3-dimensional small covers by using two operations called a connected sum and a surgery. These operations correspondent to combinatorial operations on $(Z_2)^3$-colored simple convex polytopes. We shall show that each 3-dimensional small cover can be constructed from $T^3$, $…

What makes a neural code convex?

Neural codes allow the brain to represent, process, and store information about the world. Combinatorial codes, comprised of binary patterns of neural activity, encode information via the collective behavior of populations of neurons. A code is called convex if its codewords correspond to regions defined by an arrangement of convex open sets in Euclidean space. Convex codes have been observed experimentally in many brain areas, including sensory cortices and the hippocampus, where neurons exhibit convex receptive fields. What makes a neural code convex? That is, how can we tell from the intrin…

Log-balanced combinatorial sequences

We consider log-convex sequences that satisfy an additional constraint imposed on their rate of growth. We call such sequences log-balanced. It is shown that all such sequences satisfy a pair of double inequalities. Sufficient conditions for log-balancedness are given for the case when the sequence satisfies a two- (or more-) term linear recurrence. It is shown that many combinatorially interesting sequences belong to this class, and, as a consequence, that the above-mentioned double inequalities are valid for all of them.

论Arthur迹公式的组合学、凸多面体与环面簇

On Combinatorics of the Arthur Trace Formula, Convex Polytopes, and Toric Varieties

本文从扇与多面体出发构造组合截断函数，证明其积分的收敛性与多项式性，推广了Arthur迹公式、Ehrhart多项式及Langlands引理，并给出环面簇上的几何解释。

Strict linear prices in non-convex European day-ahead electricity markets

The European power grid can be divided into several market areas where the price of electricity is determined in a day-ahead auction. Market participants can provide continuous hourly bid curves and combinatorial bids with associated quantities given the prices. The goal of our auction is to maximize the economic surplus of all participants subject to quantity constraints and price constraints. The price constraints ensure that no one incurs a loss. Only traders who submitted a combinatorial bid might miss a not-realized profit. The resulting problem is a large scale mathematical program with …

有限与可数无穷凸格从点配置出发

Finite versus uncountable convex lattices from point configurations

研究由有限点集生成的最小凸格结构，引入点配置概念，证明当n≥6时，凸格数量为不可数无穷。

Zeta Functions and the Log-behavior of Combinatorial Sequences

In this paper, we use the Riemann zeta function $ζ(x)$ and the Bessel zeta function $ζ_μ(x)$ to study the log-behavior of combinatorial sequences. We prove that $ζ(x)$ is log-convex for $x>1$. As a consequence, we deduce that the sequence $\{|B_{2n}|/(2n)!\}_{n\geq 1}$ is log-convex, where $B_n$ is the $n$-th Bernoulli number. We introduce the function $θ(x)=(2ζ(x)Γ(x+1))^{\frac{1}{x}}$, where $Γ(x)$ is the gamma function, and we show that $\log θ(x)$ is strictly increasing for $x\geq 6$. This confirms a conjecture of Sun stating that the sequence $\{\sqrt[n] {|B_{2n}}|\}_{n\geq 1}$ is strictl…

Asymptotic $r$-log-convexity and P-recursive sequences

A sequence $\{ a_n \}_{n \ge 0}$ is said to be asymptotically $r$-log-convex if it is $r$-log-convex for $n$ sufficiently large. We present a criterion on the asymptotical $r$-log-convexity based on the asymptotic behavior of $a_n a_{n+2}/a_{n+1}^2$. As an application, we show that most P-recursive sequences are asymptotic $r$-log-convexity for any integer $r$ once they are log-convex. Moreover, for a concrete integer $r$, we present a systematic method to find the explicit integer $N$ such that a P-recursive sequence $\{a_n\}_{n \ge N}$ is $r$-log-convex. This enable us to prove the $r$-log-c…

Depth with respect to a family of convex sets

We propose a notion of depth with respect to a finite family $\mathcal{F}$ of convex sets in $\mathbb{R}^d$ which we call $\text{dep}_\mathcal{F}$. We begin showing that $\text{dep}_\mathcal{F}$ satisfies some expected properties for a measure of depth and that this definition is closely related to the notion of depth proposed by J. Tukey. We show that some properties of Tukey depth extend to $\text{dep}_\mathcal{F}$ and we point out some key differences. We then focus on the following centerpoint-type question: what is the best depth $α_{d,k}$ that we can guarantee under the hypothesis that t…

The Number of Convex Polyominoes with Given Height and Width

We give a new combinatorial proof for the number of convex polyominoes whose minimum enclosing rectangle has given dimensions. We also count the subclass of these polyominoes that contain the lower left corner of the enclosing rectangle (directed polyominoes). We indicate how to sample random polyominoes in these classes. As a side result, we calculate the first and second moments of the number of common points of two monotone lattice paths between two given points.

Common tangents to convex bodies

It is well-known since the time of the Greeks that two disjoint circles in the plane have four common tangent lines. Cappell et al. proved a generalization of this fact for properly separated strictly convex bodies in higher dimensions. We have shown that the same generalization applies for arbitrary convex bodies. When the number of convex sets involved is equal to the dimension, we obtain an alternative combinatorial proof of Bisztriczky's theorem on the number of common tangents to $d$ separated convex bodies in $\Rr^d$.

A Notion of Total Dual Integrality for Convex, Semidefinite, and Extended Formulations

Total dual integrality is a powerful and unifying concept in polyhedral combinatorics and integer programming that enables the refinement of geometric min-max relations given by linear programming Strong Duality into combinatorial min-max theorems. The definition of total dual integrality (TDI) revolves around the existence of optimal dual solutions that are integral, and thus naturally applies to a host of combinatorial optimization problems that are cast as integer programs whose LP relaxations have the TDIness property. However, when combinatorial problems are formulated using more general …

An Axiomatic Duality Framework for the Theta Body and Related Convex Corners

Lovasz theta function and the related theta body of graphs have been in the center of the intersection of four research areas: combinatorial optimization, graph theory, information theory, and semidefinite optimization. In this paper, utilizing a modern convex optimization viewpoint, we provide a set of minimal conditions (axioms) under which certain key, desired properties are generalized, including the main equivalent characterizations of the theta function, the theta body of graphs, and the corresponding antiblocking duality relations. Our framework describes several semidefinite and polyhe…

包含k-平面的凸集交的Helly定理与Radon定理

Helly and Radon theorems for convex intersections containing $k$-flats

该研究针对R^d中交包含k维仿射空间的凸集族，推广了多项组合几何核心定理，k=0时可得到对应点版本的新证明，丰富了凸集族相关的组合几何理论。

内接于球面的9顶点最大体积多面体

The maximum volume polytope with nine vertices inscribed in the sphere

该研究解决了凸几何中9顶点内接球面最大体积多面体的经典问题，结合组合与几何方法，确定了其体积上界及对应构型。

A bound on a convexity measure for point sets

A planar point set is in convex position precisely when it has a convex polygonization, that is, a polygonization with maximum interior angle measure at most π. We can thus talk about the convexity of a set of points in terms of the minimum, taken over all polygonizations, of the maximum interior angle. The main result presented here is a nontrivial combinatorial upper bound of this min-max value in terms of the number of points in the set. Motivated by a particular construction, we also pose a natural conjecture for the best upper bound.

The convex hull of a Banach-Saks set

A subset $A$ of a Banach space is called Banach-Saks when every sequence in $A$ has a Ces{à}ro convergent subsequence. Our interest here focusses on the following problem: is the convex hull of a Banach-Saks set again Banach-Saks? By means of a combinatorial argument, we show that in general the answer is negative. However, sufficient conditions are given in order to obtain a positive result.

组合神经码的覆盖关系

Covering Relations in the Poset of Combinatorial Neural Codes

本文研究了组合神经码的覆盖关系，探讨了凸神经码的实现问题，并提出了关于凸神经码与多面体凸神经码等价性的猜想。

Neural codes, decidability, and a new local obstruction to convexity

Given an intersection pattern of arbitrary sets in Euclidean space, is there an arrangement of convex open sets in Euclidean space that exhibits the same intersections? This question is combinatorial and topological in nature, but is motivated by neuroscience. Specifically, we are interested in a type of neuron called a place cell, which fires precisely when an organism is in a certain region, usually convex, called a place field. The earlier question, therefore, can be rephrased as follows: Which neural codes, that is, patterns of neural activity, can arise from a collection of convex open se…