An example showing the increased power of <a href="https://en.wikipedia.org/wiki/Blichfeldt%27s_theorem" class="extiw" title="en:Blichfeldt's theorem">Blichfeldt's theorem</a> over <a href="https://en.wikipedia.org/wiki/Minkowski%27s_theorem" class="extiw" title="en:Minkowski's theorem">Minkowski's theorem</a> for finding lattice points in non-convex sets. The (closed) yellow set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>Y</mi></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle Y}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span> on the left has area 1, so by Blichfeldt's theorem it can be translated to cover two points of any lattice whose fundamental region has volume 1, such as the red lattice. It follows that the blue set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=Y-Y}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>X</mi><mo>=</mo><mi>Y</mi><mo>−<!-- − --></mo><mi>Y</mi></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle X=Y-Y}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0f72a453f2e554eef2974c25803852ef2dd820e" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.505ex; width:11.465ex; height:2.343ex;" alt="{\displaystyle X=Y-Y}"></span> on the left, the set of differences of pairs of points in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>Y</mi></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle Y}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span>, when centered on a lattice point, also contains at least one other lattice point as well as the center. In contrast, the blue rectangle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>R</mi></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle R}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> on the right, the largest convex subset of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>X</mi></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle X}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>, has too small an area for Minkowski's theorem to guarantee that it contains another lattice point, and the smaller yellow rectangle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}R}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="false" scriptlevel="0"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow><mi>R</mi></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}R}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/613af27d9f990f2f1c18ce934e0786f06b3c5a2c" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.171ex; width:3.422ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}R}"></span> within it is too small to apply Blichfeldt's theorem.
<a href="//commons.wikimedia.org/wiki/User:David_Eppstein" class="mw-redirect" title="User:David Eppstein">David Eppstein</a>
<span class="int-own-work" lang="en">Own work</span>