<p>A <i>d</i>-dimensional tensegrity framework (<i>T</i>,&#xa0;<i>p</i>) is an edge-labeled geometric graph in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {R}^d\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation>, which consists of a graph <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(T=(V,B\cup C\cup S)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>V</mi> <mo>,</mo> <mi>B</mi> <mo>∪</mo> <mi>C</mi> <mo>∪</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and a map <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(p:V\rightarrow \mathbb {R}^d\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>:</mo> <mi>V</mi> <mo stretchy="false">→</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>. The labels determine whether an edge <i>uv</i> of <i>T</i> corresponds to a fixed length bar in (<i>T</i>,&#xa0;<i>p</i>), or a cable which cannot increase in length, or a strut which cannot decrease in length. We consider minimally infinitesimally rigid <i>d</i>-dimensional tensegrity frameworks and provide tight upper bounds on the number of its edges, in terms of the number of vertices and the dimension <i>d</i>. We obtain stronger upper bounds in the case when there are no bars and the framework is in generic position. The proofs use methods from convex geometry and matroid theory. A special case of our results confirms a conjecture of Whiteley from 1987. We also give an affirmative answer to a conjecture concerning the number of edges of a graph whose three-dimensional rigidity matroid is minimally connected.</p>