Geometry

2205 Submissions

[2] viXra:2205.0055 [pdf] submitted on 2022-05-09 20:26:35

The Ehrhart Volume Conjecture Is False in Sufficiently Higher Dimensions in $\mathbb{R}^n$

Authors: Theophilus Agama
Comments: 7 Pages.

Using the method of compression, we show that volume $Vol(K)$ of a ball $K$ in $\mathbb{R}^n$ with a single lattice point in it's interior as center of mass satisfies the lower bound \begin{align} Vol(K)\gg \frac{n^n}{\sqrt{n}}\nonumber \end{align}thereby disproving the Ehrhart volume conjecture, which claims that the upper bound must hold \begin{align} Vol(K) \leq \frac{(n+1)^n}{n!}\nonumber \end{align}for all convex bodies with the required property.
Category: Geometry

[1] viXra:2205.0019 [pdf] submitted on 2022-05-02 20:43:02

On the Average Number of Integer Powered Distances in $\mathbb{r}^k$

Authors: Theophilus Agama
Comments: 7 Pages.

Using the method of compression we obtain a lower bound for the average number of $d^r$-unit distances that can be formed from a set of $n$ points in the euclidean space $\mathbb{R}^k$. By letting $\mathcal{D}_{n,d^r}$ denotes the number of $d^r$-unit distances~($r>1$~fixed) that can be formed from a set of $n$ points in $\mathbb{R}^k$, then we obtain the lower bound \begin{align} \sum \limits_{1\leq d\leq t}\mathcal{D}_{n,d^r}\gg n\sqrt[2r]{k}\log t.\nonumber \end{align}for a fixed $t>1$.
Category: Geometry